%------------------------------------------------------------------------------
% File     : ITP261^3 : TPTP v9.0.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_DeleteCorrectness 00010_000214
% 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    : 0073_VEBT_DeleteCorrectness_00010_000214 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 10531 (4574 unt;1183 typ;   0 def)
%            Number of atoms       : 29156 (11175 equ;   0 cnn)
%            Maximal formula atoms :   71 (   3 avg)
%            Number of connectives : 107864 (3005   ~; 454   |;2337   &;90159   @)
%                                         (   0 <=>;11909  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :   92 (  91 usr)
%            Number of type conns  : 4609 (4609   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1095 (1092 usr;  77 con; 0-8 aty)
%            Number of variables   : 24075 (1986   ^;21164   !; 925   ?;24075   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 07:25:35.497
%------------------------------------------------------------------------------
% Could-be-implicit typings (91)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__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_Mt__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_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__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_Mt__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_J,type,
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thf(ty_n_t__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,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__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,
    produc3843707927480180839at_nat: $tType ).

thf(ty_n_t__List__Olist_It__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_J,type,
    list_P8469869581646625389at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__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_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc7248412053542808358at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
    list_P7524865323317820941T_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    produc7272778201969148633d_enat: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    list_P5707943133018811711nt_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    produc9072475918466114483BT_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    produc4894624898956917775BT_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    produc8025551001238799321T_VEBT: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    set_Pr958786334691620121nt_int: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    list_set_nat_rat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    set_set_nat_rat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    list_list_VEBT_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    set_list_list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    set_list_VEBT_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
    set_set_list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_list_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Extended____Nat__Oenat_J_J,type,
    set_li5464603477888414924d_enat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Extended____Nat__Oenat_J_J,type,
    set_se7270636423289371942d_enat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    product_prod_nat_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    product_prod_int_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    product_prod_int_int: $tType ).

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

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

thf(ty_n_t__Option__Ooption_It__VEBT____Definitions__OVEBT_J,type,
    option_VEBT_VEBT: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    set_nat_rat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    list_list_nat: $tType ).

thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    list_VEBT_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

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

thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    list_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    set_VEBT_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    set_set_int: $tType ).

thf(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
    list_Extended_enat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    set_Extended_enat: $tType ).

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

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

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

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

thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

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

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

thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
    set_char: $tType ).

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

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

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

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

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

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

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

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

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

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

thf(ty_n_t__Code____Numeral__Ointeger,type,
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thf(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

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

thf(ty_n_t__List__Olist_I_Eo_J,type,
    list_o: $tType ).

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

thf(ty_n_t__String__Oliteral,type,
    literal: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__String__Ochar,type,
    char: $tType ).

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

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

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

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

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

% Explicit typings (1092)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

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

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

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

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

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

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

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

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

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

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

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

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

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

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

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

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

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

thf(sy_c_BNF__Def_Orel__fun_001t__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_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__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__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat_001t__Rat__Orat,type,
    bNF_re4702136315717946289at_rat: ( nat > nat > $o ) > ( rat > rat > $o ) > ( nat > rat ) > ( nat > 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_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__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_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__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).

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

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

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

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

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

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

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

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

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__Int__Oint,type,
    bit_se725231765392027082nd_int: int > int > int ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

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

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

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

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: set_nat > nat ).

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

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

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

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
    complete_Sup_Sup_int: set_int > int ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

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

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

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

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

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

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

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

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

thf(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
    condit2214826472909112428ve_nat: set_nat > $o ).

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

thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
    has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).

thf(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: int > int > product_prod_int_int > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
    unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
    unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
    unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
    unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Int__Oint,type,
    euclid3395696857347342551nt_int: int > int ).

thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Nat__Onat,type,
    euclid3398187327856392827nt_nat: nat > nat ).

thf(sy_c_Extended__Nat_OeSuc,type,
    extended_eSuc: extended_enat > extended_enat ).

thf(sy_c_Extended__Nat_Oenat,type,
    extended_enat2: nat > extended_enat ).

thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001_Eo,type,
    extended_case_enat_o: ( nat > $o ) > $o > extended_enat > $o ).

thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001t__Extended____Nat__Oenat,type,
    extend3600170679010898289d_enat: ( nat > extended_enat ) > extended_enat > extended_enat > extended_enat ).

thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nat__Oenat,type,
    extend5688581933313929465d_enat: extended_enat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
    comm_s2602460028002588243omplex: complex > nat > complex ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
    comm_s4660882817536571857er_int: int > nat > int ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
    comm_s4663373288045622133er_nat: nat > nat > nat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
    comm_s4028243227959126397er_rat: rat > nat > rat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
    comm_s7457072308508201937r_real: real > nat > real ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
    semiri5044797733671781792omplex: nat > complex ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
    semiri1406184849735516958ct_int: nat > int ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
    semiri1408675320244567234ct_nat: nat > nat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
    semiri773545260158071498ct_rat: nat > rat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
    semiri2265585572941072030t_real: nat > real ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
    invers8013647133539491842omplex: complex > complex ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
    inverse_inverse_rat: rat > rat ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: real > real ).

thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
    at_top_real: filter_real ).

thf(sy_c_Filter_Ocofinite_001t__Nat__Onat,type,
    cofinite_nat: filter_nat ).

thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
    eventually_nat: ( nat > $o ) > filter_nat > $o ).

thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
    filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
    filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
    filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).

thf(sy_c_Filter_Ofrequently_001t__Nat__Onat,type,
    frequently_nat: ( nat > $o ) > filter_nat > $o ).

thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
    prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).

thf(sy_c_Finite__Set_Ocard_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    finite_card_nat_rat: set_nat_rat > nat ).

thf(sy_c_Finite__Set_Ocard_001_Eo,type,
    finite_card_o: set_o > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
    finite_card_complex: set_complex > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Extended____Nat__Oenat,type,
    finite121521170596916366d_enat: set_Extended_enat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
    finite_card_int: set_int > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Complex__Ocomplex_J,type,
    finite5120063068150530198omplex: set_list_complex > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
    finite7441382602597825044d_enat: set_li5464603477888414924d_enat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Int__Oint_J,type,
    finite_card_list_int: set_list_int > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    finite7325466520557071688st_nat: set_list_list_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
    finite_card_list_nat: set_list_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite249151656366948015at_nat: set_li5450038453877631591at_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    finite5631907774883551598et_nat: set_list_set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    finite5915292604075114978T_VEBT: set_list_VEBT_VEBT > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite711546835091564841at_nat: set_Pr1261947904930325089at_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
    finite410649719033368117t_unit: set_Product_unit > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
    finite_card_real: set_real > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    finite8736671560171388117at_rat: set_set_nat_rat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    finite903997441450111292omplex: set_set_complex > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    finite3719263829065406702d_enat: set_se7270636423289371942d_enat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Int__Oint_J,type,
    finite_card_set_int: set_set_int > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    finite2364142230527598318st_nat: set_set_list_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
    finite_card_set_nat: set_set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    finite1149291290879098388et_nat: set_set_set_nat > nat ).

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    semila9081495762789891438tr_nat: ( nat > nat > nat ) > nat > $o ).

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

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

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    sup_su718114333110466843at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_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,
    sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

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

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

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Num__Onum,type,
    lattic8556559851467007577_o_num: ( $o > num ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Rat__Orat,type,
    lattic2140725968369957399_o_rat: ( $o > rat ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Real__Oreal,type,
    lattic8697145971487455083o_real: ( $o > real ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Num__Onum,type,
    lattic1922116423962787043ex_num: ( complex > num ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    lattic4729654577720512673ex_rat: ( complex > rat ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    lattic8794016678065449205x_real: ( complex > real ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Extended____Nat__Oenat_001t__Num__Onum,type,
    lattic402713867396545063at_num: ( extended_enat > num ) > set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Extended____Nat__Oenat_001t__Rat__Orat,type,
    lattic3210252021154270693at_rat: ( extended_enat > rat ) > set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
    lattic1189837152898106425t_real: ( extended_enat > real ) > set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Rat__Orat,type,
    lattic7811156612396918303nt_rat: ( int > rat ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
    lattic2675449441010098035t_real: ( int > real ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
    lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
    lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Num__Onum,type,
    lattic1613168225601753569al_num: ( real > num ) > set_real > real ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Rat__Orat,type,
    lattic4420706379359479199al_rat: ( real > rat ) > set_real > real ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
    lattic8440615504127631091l_real: ( real > real ) > set_real > real ).

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

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

thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
    at_infinity_real: filter_real ).

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

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

thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
    concat_nat: list_list_nat > list_nat ).

thf(sy_c_List_Oconcat_001t__VEBT____Definitions__OVEBT,type,
    concat_VEBT_VEBT: list_list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Ocount__list_001_Eo,type,
    count_list_o: list_o > $o > nat ).

thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
    count_list_int: list_int > int > nat ).

thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
    count_list_nat: list_nat > nat > nat ).

thf(sy_c_List_Ocount__list_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    count_6735058137522573441at_rat: list_set_nat_rat > set_nat_rat > nat ).

thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
    count_list_set_nat: list_set_nat > set_nat > nat ).

thf(sy_c_List_Ocount__list_001t__VEBT____Definitions__OVEBT,type,
    count_list_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > nat ).

thf(sy_c_List_Odistinct_001t__Complex__Ocomplex,type,
    distinct_complex: list_complex > $o ).

thf(sy_c_List_Odistinct_001t__Extended____Nat__Oenat,type,
    distin4523846830085650399d_enat: list_Extended_enat > $o ).

thf(sy_c_List_Odistinct_001t__Int__Oint,type,
    distinct_int: list_int > $o ).

thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
    distinct_list_nat: list_list_nat > $o ).

thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
    distinct_nat: list_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    distin6923225563576452346at_nat: list_P6011104703257516679at_nat > $o ).

thf(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
    distinct_set_nat: list_set_nat > $o ).

thf(sy_c_List_Odistinct_001t__VEBT____Definitions__OVEBT,type,
    distinct_VEBT_VEBT: list_VEBT_VEBT > $o ).

thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
    enumerate_int: nat > list_int > list_P3521021558325789923at_int ).

thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
    enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
    enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(sy_c_List_Ofind_001t__Int__Oint,type,
    find_int: ( int > $o ) > list_int > option_int ).

thf(sy_c_List_Ofind_001t__Nat__Onat,type,
    find_nat: ( nat > $o ) > list_nat > option_nat ).

thf(sy_c_List_Ofind_001t__Num__Onum,type,
    find_num: ( num > $o ) > list_num > option_num ).

thf(sy_c_List_Ofind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    find_P8199882355184865565at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > option4927543243414619207at_nat ).

thf(sy_c_List_Ofind_001t__VEBT____Definitions__OVEBT,type,
    find_VEBT_VEBT: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > option_VEBT_VEBT ).

thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
    fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).

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

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

thf(sy_c_List_Olist_OCons_001_Eo,type,
    cons_o: $o > list_o > list_o ).

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

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

thf(sy_c_List_Olist_OCons_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    cons_set_nat_rat: set_nat_rat > list_set_nat_rat > list_set_nat_rat ).

thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
    cons_set_nat: set_nat > list_set_nat > list_set_nat ).

thf(sy_c_List_Olist_OCons_001t__VEBT____Definitions__OVEBT,type,
    cons_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

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

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

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

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

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

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

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

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    set_list_VEBT_VEBT2: list_list_VEBT_VEBT > set_list_VEBT_VEBT ).

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

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

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

thf(sy_c_List_Olist_Oset_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    set_set_nat_rat2: list_set_nat_rat > set_set_nat_rat ).

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

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

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

thf(sy_c_List_Olist__update_001_Eo,type,
    list_update_o: list_o > nat > $o > list_o ).

thf(sy_c_List_Olist__update_001t__Int__Oint,type,
    list_update_int: list_int > nat > int > list_int ).

thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
    list_update_nat: list_nat > nat > nat > list_nat ).

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

thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
    list_update_real: list_real > nat > real > list_real ).

thf(sy_c_List_Olist__update_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    list_u886106648575569423at_rat: list_set_nat_rat > nat > set_nat_rat > list_set_nat_rat ).

thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
    list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).

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

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

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

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_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__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr3474266648193625910T_VEBT: list_P7524865323317820941T_VEBT > nat > produc1531783533982839933T_VEBT ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).

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

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

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_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).

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

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

thf(sy_c_List_Onth_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    nth_set_nat_rat: list_set_nat_rat > nat > set_nat_rat ).

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

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

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__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

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__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

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_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

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

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

thf(sy_c_List_OremoveAll_001_Eo,type,
    removeAll_o: $o > list_o > list_o ).

thf(sy_c_List_OremoveAll_001t__Int__Oint,type,
    removeAll_int: int > list_int > list_int ).

thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
    removeAll_nat: nat > list_nat > list_nat ).

thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    remove3673390508374433037at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_OremoveAll_001t__Real__Oreal,type,
    removeAll_real: real > list_real > list_real ).

thf(sy_c_List_OremoveAll_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    remove939820145577552881at_rat: set_nat_rat > list_set_nat_rat > list_set_nat_rat ).

thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Nat__Onat_J,type,
    removeAll_set_nat: set_nat > list_set_nat > list_set_nat ).

thf(sy_c_List_OremoveAll_001t__VEBT____Definitions__OVEBT,type,
    removeAll_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).

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

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_Orotate1_001t__VEBT____Definitions__OVEBT,type,
    rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).

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

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

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

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

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

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

thf(sy_c_List_Oupto__rel,type,
    upto_rel: product_prod_int_int > product_prod_int_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__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

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__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

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_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

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

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

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_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: nat > code_integer ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat ).

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_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_It__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    size_s3959913991096427681at_rat: list_set_nat_rat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
    size_size_option_nat: option_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
    size_size_option_num: option_num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Num_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__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: complex > complex ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Num_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__Complex__Ocomplex,type,
    numera6690914467698888265omplex: num > complex ).

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

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

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

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

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

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

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

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

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: int > option_int ).

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

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

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

thf(sy_c_Option_Ooption_OSome_001t__VEBT____Definitions__OVEBT,type,
    some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).

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

thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
    size_option_nat: ( nat > nat ) > option_nat > nat ).

thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
    size_option_num: ( num > nat ) > option_num > nat ).

thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).

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

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_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_Eo_J,type,
    bot_bot_nat_rat_o: ( nat > rat ) > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
    bot_bot_o_o: $o > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    bot_bot_int_int_o: int > int > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
    bot_bot_int_o: int > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bot_bot_nat_nat_o: nat > 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_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    bot_bo4898103413517107610_nat_o: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
    bot_bot_real_o: real > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J_M_Eo_J,type,
    bot_bo3445895781125589758_rat_o: set_nat_rat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    bot_bo394778441745866138_nat_o: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__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_M_062_It__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_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    bot_bot_filter_nat: filter_nat ).

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_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    bot_bot_set_nat_rat: set_nat_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
    bot_bot_set_o: set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    bot_bo7653980558646680370d_enat: set_Extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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_J,type,
    bot_bo5327735625951526323at_nat: set_Pr8693737435421807431at_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__Oprod_It__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_Mt__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_J_J,type,
    bot_bo4948859079157340979at_nat: set_Pr7459493094073627847at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    bot_bo6797373522285170759at_rat: set_set_nat_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    bot_bot_set_set_int: set_set_int ).

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__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_M_Eo_J,type,
    ord_less_o_o: ( $o > $o ) > ( $o > $o ) > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J_M_Eo_J,type,
    ord_le6823063569548456766_rat_o: ( set_nat_rat > $o ) > ( set_nat_rat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
    ord_less_o: $o > $o > $o ).

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
    ord_less_set_o: set_o > set_o > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    ord_le1311537459589289991at_rat: set_set_nat_rat > set_set_nat_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_less_set_set_int: set_set_int > set_set_int > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
    ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
    ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
    ord_le6558929396352911974_nat_o: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    ord_le1598226405681992910_int_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    ord_le8369615600986905444_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le5604493270027003598_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le1077754993875142464_nat_o: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le7812727212727832188_nat_o: ( produc9072475918466114483BT_nat > $o ) > ( produc9072475918466114483BT_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J_M_Eo_J,type,
    ord_le4100815579384348210_rat_o: ( set_nat_rat > $o ) > ( set_nat_rat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    ord_le3935385432712749774_nat_o: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__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_M_062_It__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_M_Eo_J_J,type,
    ord_le3072208448688395470_nat_o: ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ) > ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
    ord_less_eq_o: $o > $o > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
    ord_le4104064031414453916r_real: filter_real > filter_real > $o ).

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    ord_le2679597024174929757at_rat: set_nat_rat > set_nat_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
    ord_less_eq_set_o: set_o > set_o > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_J,type,
    ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_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,
    ord_le1268244103169919719at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__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_Mt__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_J_J,type,
    ord_le5997549366648089703at_nat: set_Pr7459493094073627847at_nat > set_Pr7459493094073627847at_nat > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J_J,type,
    ord_le4375437777232675859at_rat: set_set_nat_rat > set_set_nat_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
    ord_max_Code_integer: code_integer > code_integer > code_integer ).

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

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

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

thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: num > num > num ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: rat > rat > rat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: real > real > real ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_I_Eo_J,type,
    ord_max_set_o: set_o > set_o > set_o ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: set_int > set_int > set_int ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_max_set_real: set_real > set_real > set_real ).

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

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

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

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

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

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

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

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

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

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

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

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

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

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

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

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

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

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
    product_Pair_int_nat: int > nat > product_prod_int_nat ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
    product_Pair_nat_int: nat > int > product_prod_nat_int ).

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

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

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).

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

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).

thf(sy_c_Product__Type_OPair_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_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,
    produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat ).

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

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
    produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).

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

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
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thf(sy_c_Product__Type_Ounit_ORep__unit,type,
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thf(sy_c_Quotient_Oquot__type_Oabs_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
    quot_a3129823074075660125t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( set_nat_rat > real ) > ( nat > rat ) > real ).

thf(sy_c_Quotient_Oquot__type_Orep_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
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thf(sy_c_Rat_OFract,type,
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thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
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thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
    field_7254667332652039916t_real: rat > real ).

thf(sy_c_Rat_Oquotient__of,type,
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thf(sy_c_Real_OReal,type,
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thf(sy_c_Real_Ocauchy,type,
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thf(sy_c_Real_Ocr__real,type,
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thf(sy_c_Real_Opcr__real,type,
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thf(sy_c_Real_Opositive,type,
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thf(sy_c_Real_Oreal_OAbs__real,type,
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thf(sy_c_Real_Oreal_ORep__real,type,
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thf(sy_c_Real_Orealrel,type,
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thf(sy_c_Real_Orep__real,type,
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thf(sy_c_Real_Ovanishes,type,
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thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
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thf(sy_c_Relation_ODomainp_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,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_Relation_Orelcompp_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
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thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Relation_Otrans_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Relation_Otransp_001_062_It__Nat__Onat_Mt__Rat__Orat_J,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__Complex__Ocomplex,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_Odivide__class_Odivide_001t__Real__Oreal,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_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,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_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_Series_Osuminf_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
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thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osummable_001t__Int__Oint,type,
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thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
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thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
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thf(sy_c_Series_Osums_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
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thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,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__Complex__Ocomplex_J,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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

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

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

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( int > nat ) > set_int > set_nat ).

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

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

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

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

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

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

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

thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    insert_nat_rat: ( nat > rat ) > set_nat_rat > set_nat_rat ).

thf(sy_c_Set_Oinsert_001_Eo,type,
    insert_o: $o > set_o > set_o ).

thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
    insert_complex: complex > set_complex > set_complex ).

thf(sy_c_Set_Oinsert_001t__Extended____Nat__Oenat,type,
    insert_Extended_enat: extended_enat > set_Extended_enat > set_Extended_enat ).

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

thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
    insert_list_nat: list_nat > set_list_nat > set_list_nat ).

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

thf(sy_c_Set_Oinsert_001t__Num__Onum,type,
    insert_num: num > set_num > set_num ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

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,
    insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
    insert_rat: rat > set_rat > set_rat ).

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

thf(sy_c_Set_Oinsert_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    insert_set_nat_rat: set_nat_rat > set_set_nat_rat > set_set_nat_rat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
    insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_Set_Ois__empty_001_Eo,type,
    is_empty_o: set_o > $o ).

thf(sy_c_Set_Ois__empty_001t__Int__Oint,type,
    is_empty_int: set_int > $o ).

thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
    is_empty_nat: set_nat > $o ).

thf(sy_c_Set_Ois__empty_001t__Real__Oreal,type,
    is_empty_real: set_real > $o ).

thf(sy_c_Set_Ois__singleton_001_Eo,type,
    is_singleton_o: set_o > $o ).

thf(sy_c_Set_Ois__singleton_001t__Complex__Ocomplex,type,
    is_singleton_complex: set_complex > $o ).

thf(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
    is_singleton_int: set_int > $o ).

thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
    is_sin2641923865335537900st_nat: set_list_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
    is_singleton_nat: set_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    is_sin2850979758926227957at_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
    is_singleton_real: set_real > $o ).

thf(sy_c_Set_Ois__singleton_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    is_sin2571591796506819849at_rat: set_set_nat_rat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
    is_singleton_set_nat: set_set_nat > $o ).

thf(sy_c_Set_Oremove_001_Eo,type,
    remove_o: $o > set_o > set_o ).

thf(sy_c_Set_Oremove_001t__Int__Oint,type,
    remove_int: int > set_int > set_int ).

thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
    remove_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    remove6466555014256735590at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oremove_001t__Real__Oreal,type,
    remove_real: real > set_real > set_real ).

thf(sy_c_Set_Oremove_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    remove_set_nat_rat: set_nat_rat > set_set_nat_rat > set_set_nat_rat ).

thf(sy_c_Set_Oremove_001t__Set__Oset_It__Nat__Onat_J,type,
    remove_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Set_Othe__elem_001_Eo,type,
    the_elem_o: set_o > $o ).

thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
    the_elem_int: set_int > int ).

thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
    the_elem_nat: set_nat > nat ).

thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_el2281957884133575798at_nat: set_Pr1261947904930325089at_nat > product_prod_nat_nat ).

thf(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
    the_elem_real: set_real > real ).

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

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_Eo,type,
    set_or8904488021354931149Most_o: $o > $o > set_o ).

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    set_or5795412311047298440at_rat: set_nat_rat > set_nat_rat > set_set_nat_rat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or370866239135849197et_int: set_int > set_int > set_set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

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

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

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

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

thf(sy_c_Transfer_Oleft__total_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    left_t2768085380646472630at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).

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

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

thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Ounit_001_Eo,type,
    type_d6188575255521822967unit_o: ( product_unit > $o ) > ( $o > product_unit ) > set_o > $o ).

thf(sy_c_Typedef_Otype__definition_001t__Real__Oreal_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    type_d8072115097938612567at_rat: ( real > set_nat_rat ) > ( set_nat_rat > real ) > set_set_nat_rat > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

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

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

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

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

thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > nat > nat > $o ).

thf(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).

thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).

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

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__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_fChoice_001t__Real__Oreal,type,
    fChoice_real: ( real > $o ) > real ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    member_nat_rat: ( nat > rat ) > set_nat_rat > $o ).

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

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

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

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

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

thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $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__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $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__Oprod_It__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_Mt__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_J,type,
    member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    member_set_nat_rat: set_nat_rat > set_set_nat_rat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: set_int > set_set_int > $o ).

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

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

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_t,type,
    t: vEBT_VEBT ).

thf(sy_v_x,type,
    x: nat ).

% Relevant facts (9308)
thf(fact_0_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_1_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_2_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_3_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_4_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_5_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_6_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( Deg = N ) ) ).

% deg_deg_n
thf(fact_7_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_8_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_9_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_10_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% deg_not_0
thf(fact_11_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_12_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_13_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_14_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ Xs )
              & ( ord_less_nat @ X2 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_15_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ Xs )
              & ( ord_less_nat @ A @ X2 ) ) ) ) ).

% succ_none_empty
thf(fact_16_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).

% buildup_gives_valid
thf(fact_17_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_18_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_19_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_20_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_21_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B ) ) ).

% vebt_delete.simps(1)
thf(fact_22_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A2: set_nat] :
      ( ( ord_less_nat @ Z @ X )
     => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ A2 )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_23_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ B2 )
         => ( ( A2 = B2 )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_24_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X22 )
        = ( vEBT_Leaf @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% VEBT.inject(2)
thf(fact_25_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
thf(fact_26_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [A3: set_nat] :
            ( ( finite_finite_nat @ A3 )
           => ( ! [B3: set_nat] :
                  ( ( ord_less_set_nat @ B3 @ A3 )
                 => ( P @ B3 ) )
             => ( P @ A3 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_27_finite__psubset__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ! [A3: set_int] :
            ( ( finite_finite_int @ A3 )
           => ( ! [B3: set_int] :
                  ( ( ord_less_set_int @ B3 @ A3 )
                 => ( P @ B3 ) )
             => ( P @ A3 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_28_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [A3: set_complex] :
            ( ( finite3207457112153483333omplex @ A3 )
           => ( ! [B3: set_complex] :
                  ( ( ord_less_set_complex @ B3 @ A3 )
                 => ( P @ B3 ) )
             => ( P @ A3 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_29_finite__psubset__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ! [A3: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ A3 )
           => ( ! [B3: set_Pr1261947904930325089at_nat] :
                  ( ( ord_le7866589430770878221at_nat @ B3 @ A3 )
                 => ( P @ B3 ) )
             => ( P @ A3 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_30_finite__psubset__induct,axiom,
    ! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [A3: set_Extended_enat] :
            ( ( finite4001608067531595151d_enat @ A3 )
           => ( ! [B3: set_Extended_enat] :
                  ( ( ord_le2529575680413868914d_enat @ B3 @ A3 )
                 => ( P @ B3 ) )
             => ( P @ A3 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_31_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_32_zero__reorient,axiom,
    ! [X: literal] :
      ( ( zero_zero_literal = X )
      = ( X = zero_zero_literal ) ) ).

% zero_reorient
thf(fact_33_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_34_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_35_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_36_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_37_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_38_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M: nat] :
              ( ( ord_less_nat @ M @ N2 )
              & ~ ( P @ M ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_39_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M: nat] :
              ( ( ord_less_nat @ M @ N2 )
             => ( P @ M ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_40_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_41_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_42_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_43_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_44_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
      = ( ( ord_less_nat @ M2 @ N )
        | ( ord_less_nat @ N @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_45_mem__Collect__eq,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( member_o @ A @ ( collect_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_46_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_47_mem__Collect__eq,axiom,
    ! [A: set_nat_rat,P: set_nat_rat > $o] :
      ( ( member_set_nat_rat @ A @ ( collect_set_nat_rat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
    ! [A: nat > rat,P: ( nat > rat ) > $o] :
      ( ( member_nat_rat @ A @ ( collect_nat_rat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_51_Collect__mem__eq,axiom,
    ! [A2: set_o] :
      ( ( collect_o
        @ ^ [X3: $o] : ( member_o @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
    ! [A2: set_set_nat_rat] :
      ( ( collect_set_nat_rat
        @ ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
    ! [A2: set_nat_rat] :
      ( ( collect_nat_rat
        @ ^ [X3: nat > rat] : ( member_nat_rat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_57_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X4: set_nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_58_Collect__cong,axiom,
    ! [P: set_nat_rat > $o,Q: set_nat_rat > $o] :
      ( ! [X4: set_nat_rat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_set_nat_rat @ P )
        = ( collect_set_nat_rat @ Q ) ) ) ).

% Collect_cong
thf(fact_59_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_60_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_61_Collect__cong,axiom,
    ! [P: ( nat > rat ) > $o,Q: ( nat > rat ) > $o] :
      ( ! [X4: nat > rat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_nat_rat @ P )
        = ( collect_nat_rat @ Q ) ) ) ).

% Collect_cong
thf(fact_62_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_63_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_64_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_65_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_66_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M: nat] :
                  ( ( ord_less_nat @ M @ N2 )
                  & ~ ( P @ M ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_67_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_68_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_69_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_70_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_71_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_72_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_73_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X222: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X222 ) ) ) ).

% VEBT.exhaust
thf(fact_74_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X22 ) ) ).

% VEBT.distinct(1)
thf(fact_75_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X @ Y )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ X @ Z2 ) )
           => ( ord_less_eq_nat @ Y @ Z2 ) ) ) ) ).

% succ_member
thf(fact_76_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ Z2 @ X ) )
           => ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ).

% pred_member
thf(fact_77_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_78_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_79_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_80_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A4: $o,B4: $o] :
            ( T
            = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).

% deg1Leaf
thf(fact_81_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A5: $o,B5: $o] :
          ( T
          = ( vEBT_Leaf @ A5 @ B5 ) ) ) ).

% deg_1_Leaf
thf(fact_82_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( N = one_one_nat )
       => ? [A5: $o,B5: $o] :
            ( T
            = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).

% deg_1_Leafy
thf(fact_83_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N3: set_nat] :
        ? [M3: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N3 )
         => ( ord_less_nat @ X3 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_84_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M3: nat] :
          ? [N4: nat] :
            ( ( ord_less_nat @ M3 @ N4 )
            & ( member_nat @ N4 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_85_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ N5 )
         => ( ord_less_nat @ X4 @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% bounded_nat_set_is_finite
thf(fact_86_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M4: nat] :
          ( ( ord_less_nat @ K @ M4 )
         => ? [N6: nat] :
              ( ( ord_less_nat @ M4 @ N6 )
              & ( member_nat @ N6 @ S2 ) ) )
     => ~ ( finite_finite_nat @ S2 ) ) ).

% unbounded_k_infinite
thf(fact_87_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs2: set_nat,X3: nat] :
          ( ( member_nat @ X3 @ Xs2 )
          & ! [Y2: nat] :
              ( ( member_nat @ Y2 @ Xs2 )
             => ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ) ).

% max_in_set_def
thf(fact_88_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs2: set_nat,X3: nat] :
          ( ( member_nat @ X3 @ Xs2 )
          & ! [Y2: nat] :
              ( ( member_nat @ Y2 @ Xs2 )
             => ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).

% min_in_set_def
thf(fact_89_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X3: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X3 )
          | ( vEBT_VEBT_membermima @ T2 @ X3 ) ) ) ) ).

% both_member_options_def
thf(fact_90_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_91_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_92_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_93_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_94_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_95_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y4: nat] :
                ( ( P @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_96_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
      = ( X = one_one_complex ) ) ).

% one_reorient
thf(fact_97_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_98_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_99_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_100_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_101_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
      | ( ord_less_eq_nat @ N @ M2 ) ) ).

% nat_le_linear
thf(fact_102_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% le_antisym
thf(fact_103_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_104_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_105_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_106_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M5: nat] :
      ( ( P @ X )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M5 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X2: nat] :
                    ( ( P @ X2 )
                   => ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_107_finite__has__minimal,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A2 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_108_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_109_finite__has__minimal,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ? [X4: $o] :
            ( ( member_o @ X4 @ A2 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A2 )
               => ( ( ord_less_eq_o @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_110_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( ( member_set_int @ X4 @ A2 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_111_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_112_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_113_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_114_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_115_finite__has__maximal,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A2 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_116_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_117_finite__has__maximal,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ? [X4: $o] :
            ( ( member_o @ X4 @ A2 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A2 )
               => ( ( ord_less_eq_o @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_118_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( ( member_set_int @ X4 @ A2 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_119_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_120_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_121_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_122_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_123_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: set_nat > set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ! [X4: set_nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: set_nat,Y3: set_nat,Z3: set_nat] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: set_nat] :
                ( ( member_set_nat @ X4 @ A2 )
               => ? [Y4: set_nat] :
                    ( ( member_set_nat @ Y4 @ A2 )
                    & ( R @ X4 @ Y4 ) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_124_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat_rat,R: set_nat_rat > set_nat_rat > $o] :
      ( ( finite6430367030675640852at_rat @ A2 )
     => ( ! [X4: set_nat_rat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: set_nat_rat,Y3: set_nat_rat,Z3: set_nat_rat] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: set_nat_rat] :
                ( ( member_set_nat_rat @ X4 @ A2 )
               => ? [Y4: set_nat_rat] :
                    ( ( member_set_nat_rat @ Y4 @ A2 )
                    & ( R @ X4 @ Y4 ) ) )
           => ( A2 = bot_bo6797373522285170759at_rat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_125_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: complex > complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: complex,Y3: complex,Z3: complex] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A2 )
               => ? [Y4: complex] :
                    ( ( member_complex @ Y4 @ A2 )
                    & ( R @ X4 @ Y4 ) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_126_finite__transitivity__chain,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ! [X4: product_prod_nat_nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A2 )
               => ? [Y4: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y4 @ A2 )
                    & ( R @ X4 @ Y4 ) ) )
           => ( A2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_127_finite__transitivity__chain,axiom,
    ! [A2: set_Extended_enat,R: extended_enat > extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X4: extended_enat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: extended_enat,Y3: extended_enat,Z3: extended_enat] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ A2 )
               => ? [Y4: extended_enat] :
                    ( ( member_Extended_enat @ Y4 @ A2 )
                    & ( R @ X4 @ Y4 ) ) )
           => ( A2 = bot_bo7653980558646680370d_enat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_128_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: real > real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X4: real] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: real,Y3: real,Z3: real] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ A2 )
               => ? [Y4: real] :
                    ( ( member_real @ Y4 @ A2 )
                    & ( R @ X4 @ Y4 ) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_129_finite__transitivity__chain,axiom,
    ! [A2: set_o,R: $o > $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ! [X4: $o] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: $o,Y3: $o,Z3: $o] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ A2 )
               => ? [Y4: $o] :
                    ( ( member_o @ Y4 @ A2 )
                    & ( R @ X4 @ Y4 ) ) )
           => ( A2 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_130_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A2 )
               => ? [Y4: nat] :
                    ( ( member_nat @ Y4 @ A2 )
                    & ( R @ X4 @ Y4 ) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_131_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: int > int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: int,Y3: int,Z3: int] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ A2 )
               => ? [Y4: int] :
                    ( ( member_int @ Y4 @ A2 )
                    & ( R @ X4 @ Y4 ) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_132_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M3: nat] :
          ? [N4: nat] :
            ( ( ord_less_eq_nat @ M3 @ N4 )
            & ( member_nat @ N4 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_133_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N3: set_nat] :
        ? [M3: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N3 )
         => ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_134_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_135_finite__has__maximal2,axiom,
    ! [A2: set_o,A: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ A @ A2 )
       => ? [X4: $o] :
            ( ( member_o @ X4 @ A2 )
            & ( ord_less_eq_o @ A @ X4 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A2 )
               => ( ( ord_less_eq_o @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_136_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ A @ A2 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A2 )
            & ( ord_less_eq_set_nat @ A @ X4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_137_finite__has__maximal2,axiom,
    ! [A2: set_set_nat_rat,A: set_nat_rat] :
      ( ( finite6430367030675640852at_rat @ A2 )
     => ( ( member_set_nat_rat @ A @ A2 )
       => ? [X4: set_nat_rat] :
            ( ( member_set_nat_rat @ X4 @ A2 )
            & ( ord_le2679597024174929757at_rat @ A @ X4 )
            & ! [Xa: set_nat_rat] :
                ( ( member_set_nat_rat @ Xa @ A2 )
               => ( ( ord_le2679597024174929757at_rat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_138_finite__has__maximal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ A @ A2 )
       => ? [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A2 )
            & ( ord_le2932123472753598470d_enat @ A @ X4 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_139_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X4: set_int] :
            ( ( member_set_int @ X4 @ A2 )
            & ( ord_less_eq_set_int @ A @ X4 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_140_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A2 )
            & ( ord_less_eq_rat @ A @ X4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_141_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A2 )
            & ( ord_less_eq_num @ A @ X4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_142_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ( ord_less_eq_nat @ A @ X4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_143_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A2 )
            & ( ord_less_eq_int @ A @ X4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_144_finite__has__minimal2,axiom,
    ! [A2: set_o,A: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ A @ A2 )
       => ? [X4: $o] :
            ( ( member_o @ X4 @ A2 )
            & ( ord_less_eq_o @ X4 @ A )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A2 )
               => ( ( ord_less_eq_o @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_145_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ A @ A2 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A2 )
            & ( ord_less_eq_set_nat @ X4 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_146_finite__has__minimal2,axiom,
    ! [A2: set_set_nat_rat,A: set_nat_rat] :
      ( ( finite6430367030675640852at_rat @ A2 )
     => ( ( member_set_nat_rat @ A @ A2 )
       => ? [X4: set_nat_rat] :
            ( ( member_set_nat_rat @ X4 @ A2 )
            & ( ord_le2679597024174929757at_rat @ X4 @ A )
            & ! [Xa: set_nat_rat] :
                ( ( member_set_nat_rat @ Xa @ A2 )
               => ( ( ord_le2679597024174929757at_rat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_147_finite__has__minimal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ A @ A2 )
       => ? [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A2 )
            & ( ord_le2932123472753598470d_enat @ X4 @ A )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_148_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X4: set_int] :
            ( ( member_set_int @ X4 @ A2 )
            & ( ord_less_eq_set_int @ X4 @ A )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_149_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A2 )
            & ( ord_less_eq_rat @ X4 @ A )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_150_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A2 )
            & ( ord_less_eq_num @ X4 @ A )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_151_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ( ord_less_eq_nat @ X4 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_152_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A2 )
            & ( ord_less_eq_int @ X4 @ A )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_153_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_154_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_155_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_156_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_157_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M3 @ N4 )
          & ( M3 != N4 ) ) ) ) ).

% nat_less_le
thf(fact_158_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_159_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N4: nat] :
          ( ( ord_less_nat @ M3 @ N4 )
          | ( M3 = N4 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_160_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_161_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_162_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_163_finite_OemptyI,axiom,
    finite3207457112153483333omplex @ bot_bot_set_complex ).

% finite.emptyI
thf(fact_164_finite_OemptyI,axiom,
    finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).

% finite.emptyI
thf(fact_165_finite_OemptyI,axiom,
    finite4001608067531595151d_enat @ bot_bo7653980558646680370d_enat ).

% finite.emptyI
thf(fact_166_finite_OemptyI,axiom,
    finite_finite_real @ bot_bot_set_real ).

% finite.emptyI
thf(fact_167_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_168_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_169_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_170_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S2 )
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
thf(fact_171_infinite__imp__nonempty,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S2 )
     => ( S2 != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
thf(fact_172_infinite__imp__nonempty,axiom,
    ! [S2: set_Extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ S2 )
     => ( S2 != bot_bo7653980558646680370d_enat ) ) ).

% infinite_imp_nonempty
thf(fact_173_infinite__imp__nonempty,axiom,
    ! [S2: set_real] :
      ( ~ ( finite_finite_real @ S2 )
     => ( S2 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
thf(fact_174_infinite__imp__nonempty,axiom,
    ! [S2: set_o] :
      ( ~ ( finite_finite_o @ S2 )
     => ( S2 != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_175_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_176_infinite__imp__nonempty,axiom,
    ! [S2: set_int] :
      ( ~ ( finite_finite_int @ S2 )
     => ( S2 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_177_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_178_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).

% VEBT_internal.membermima.simps(1)
thf(fact_179_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K2 )
               => ~ ( P @ I3 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_180_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_181_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_182_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_183_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs2: set_nat,X3: nat,Y2: nat] :
          ( ( member_nat @ Y2 @ Xs2 )
          & ( ord_less_nat @ Y2 @ X3 )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs2 )
             => ( ( ord_less_nat @ Z2 @ X3 )
               => ( ord_less_eq_nat @ Z2 @ Y2 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_184_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs2: set_nat,X3: nat,Y2: nat] :
          ( ( member_nat @ Y2 @ Xs2 )
          & ( ord_less_nat @ X3 @ Y2 )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs2 )
             => ( ( ord_less_nat @ X3 @ Z2 )
               => ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_185_infinite__growing,axiom,
    ! [X5: set_Extended_enat] :
      ( ( X5 != bot_bo7653980558646680370d_enat )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ X5 )
           => ? [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ X5 )
                & ( ord_le72135733267957522d_enat @ X4 @ Xa ) ) )
       => ~ ( finite4001608067531595151d_enat @ X5 ) ) ) ).

% infinite_growing
thf(fact_186_infinite__growing,axiom,
    ! [X5: set_o] :
      ( ( X5 != bot_bot_set_o )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ X5 )
           => ? [Xa: $o] :
                ( ( member_o @ Xa @ X5 )
                & ( ord_less_o @ X4 @ Xa ) ) )
       => ~ ( finite_finite_o @ X5 ) ) ) ).

% infinite_growing
thf(fact_187_infinite__growing,axiom,
    ! [X5: set_real] :
      ( ( X5 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ X5 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X5 )
                & ( ord_less_real @ X4 @ Xa ) ) )
       => ~ ( finite_finite_real @ X5 ) ) ) ).

% infinite_growing
thf(fact_188_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X4: rat] :
            ( ( member_rat @ X4 @ X5 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X5 )
                & ( ord_less_rat @ X4 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X5 ) ) ) ).

% infinite_growing
thf(fact_189_infinite__growing,axiom,
    ! [X5: set_num] :
      ( ( X5 != bot_bot_set_num )
     => ( ! [X4: num] :
            ( ( member_num @ X4 @ X5 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X5 )
                & ( ord_less_num @ X4 @ Xa ) ) )
       => ~ ( finite_finite_num @ X5 ) ) ) ).

% infinite_growing
thf(fact_190_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X5 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X5 )
                & ( ord_less_nat @ X4 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X5 ) ) ) ).

% infinite_growing
thf(fact_191_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ X5 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X5 )
                & ( ord_less_int @ X4 @ Xa ) ) )
       => ~ ( finite_finite_int @ X5 ) ) ) ).

% infinite_growing
thf(fact_192_ex__min__if__finite,axiom,
    ! [S2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ S2 )
            & ~ ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ S2 )
                  & ( ord_le72135733267957522d_enat @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_193_ex__min__if__finite,axiom,
    ! [S2: set_o] :
      ( ( finite_finite_o @ S2 )
     => ( ( S2 != bot_bot_set_o )
       => ? [X4: $o] :
            ( ( member_o @ X4 @ S2 )
            & ~ ? [Xa: $o] :
                  ( ( member_o @ Xa @ S2 )
                  & ( ord_less_o @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_194_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ S2 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S2 )
                  & ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_195_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( ( finite_finite_rat @ S2 )
     => ( ( S2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ S2 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S2 )
                  & ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_196_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( ( finite_finite_num @ S2 )
     => ( ( S2 != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ S2 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S2 )
                  & ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_197_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ S2 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S2 )
                  & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_198_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ S2 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S2 )
                  & ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_199_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_200_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_201_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_202_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_203_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_204_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_205_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_206_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_207_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_208_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_209_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_210_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_211_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% zero_less_one_class.zero_le_one
thf(fact_212_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_213_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_214_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_215_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_216_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_217_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_218_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_219_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_220_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_221_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_222_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_223_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_224_empty__iff,axiom,
    ! [C: set_nat_rat] :
      ~ ( member_set_nat_rat @ C @ bot_bo6797373522285170759at_rat ) ).

% empty_iff
thf(fact_225_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_226_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_227_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_228_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_229_empty__subsetI,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% empty_subsetI
thf(fact_230_empty__subsetI,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% empty_subsetI
thf(fact_231_empty__subsetI,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% empty_subsetI
thf(fact_232_empty__subsetI,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% empty_subsetI
thf(fact_233_subset__empty,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_234_subset__empty,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_235_subset__empty,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_236_subset__empty,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_237_empty__Collect__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P ) )
      = ( ! [X3: set_nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_238_empty__Collect__eq,axiom,
    ! [P: set_nat_rat > $o] :
      ( ( bot_bo6797373522285170759at_rat
        = ( collect_set_nat_rat @ P ) )
      = ( ! [X3: set_nat_rat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_239_empty__Collect__eq,axiom,
    ! [P: ( nat > rat ) > $o] :
      ( ( bot_bot_set_nat_rat
        = ( collect_nat_rat @ P ) )
      = ( ! [X3: nat > rat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_240_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X3: real] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_241_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X3: $o] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_242_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_243_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X3: int] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_244_Collect__empty__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( ! [X3: set_nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_245_Collect__empty__eq,axiom,
    ! [P: set_nat_rat > $o] :
      ( ( ( collect_set_nat_rat @ P )
        = bot_bo6797373522285170759at_rat )
      = ( ! [X3: set_nat_rat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_246_Collect__empty__eq,axiom,
    ! [P: ( nat > rat ) > $o] :
      ( ( ( collect_nat_rat @ P )
        = bot_bot_set_nat_rat )
      = ( ! [X3: nat > rat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_247_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X3: real] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_248_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X3: $o] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_249_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_250_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X3: int] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_251_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( ! [X3: set_nat] :
            ~ ( member_set_nat @ X3 @ A2 ) )
      = ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_252_all__not__in__conv,axiom,
    ! [A2: set_set_nat_rat] :
      ( ( ! [X3: set_nat_rat] :
            ~ ( member_set_nat_rat @ X3 @ A2 ) )
      = ( A2 = bot_bo6797373522285170759at_rat ) ) ).

% all_not_in_conv
thf(fact_253_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ( ! [X3: real] :
            ~ ( member_real @ X3 @ A2 ) )
      = ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_254_all__not__in__conv,axiom,
    ! [A2: set_o] :
      ( ( ! [X3: $o] :
            ~ ( member_o @ X3 @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_255_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ! [X3: nat] :
            ~ ( member_nat @ X3 @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_256_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ! [X3: int] :
            ~ ( member_int @ X3 @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_257_psubsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_int @ A2 @ B2 ) ) ) ).

% psubsetI
thf(fact_258_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_set_int @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_259_subset__psubset__trans,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( ord_less_set_int @ B2 @ C2 )
       => ( ord_less_set_int @ A2 @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_260_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_261_psubset__subset__trans,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C2 )
       => ( ord_less_set_int @ A2 @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_262_psubset__imp__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% psubset_imp_subset
thf(fact_263_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% psubset_eq
thf(fact_264_psubsetE,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
         => ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).

% psubsetE
thf(fact_265_rev__finite__subset,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_266_rev__finite__subset,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_267_rev__finite__subset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
       => ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_268_rev__finite__subset,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_269_rev__finite__subset,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( finite_finite_int @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_270_infinite__super,axiom,
    ! [S2: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S2 @ T3 )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_271_infinite__super,axiom,
    ! [S2: set_complex,T3: set_complex] :
      ( ( ord_le211207098394363844omplex @ S2 @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S2 )
       => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).

% infinite_super
thf(fact_272_infinite__super,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S2 @ T3 )
     => ( ~ ( finite6177210948735845034at_nat @ S2 )
       => ~ ( finite6177210948735845034at_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_273_infinite__super,axiom,
    ! [S2: set_Extended_enat,T3: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
     => ( ~ ( finite4001608067531595151d_enat @ S2 )
       => ~ ( finite4001608067531595151d_enat @ T3 ) ) ) ).

% infinite_super
thf(fact_274_infinite__super,axiom,
    ! [S2: set_int,T3: set_int] :
      ( ( ord_less_eq_set_int @ S2 @ T3 )
     => ( ~ ( finite_finite_int @ S2 )
       => ~ ( finite_finite_int @ T3 ) ) ) ).

% infinite_super
thf(fact_275_finite__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( finite_finite_nat @ B2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_276_finite__subset,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_subset
thf(fact_277_finite__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ( finite6177210948735845034at_nat @ B2 )
       => ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_278_finite__subset,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% finite_subset
thf(fact_279_finite__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( finite_finite_int @ B2 )
       => ( finite_finite_int @ A2 ) ) ) ).

% finite_subset
thf(fact_280_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_281_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_282_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_283_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
      = ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_284_ex__in__conv,axiom,
    ! [A2: set_set_nat_rat] :
      ( ( ? [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ A2 ) )
      = ( A2 != bot_bo6797373522285170759at_rat ) ) ).

% ex_in_conv
thf(fact_285_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ( ? [X3: real] : ( member_real @ X3 @ A2 ) )
      = ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_286_ex__in__conv,axiom,
    ! [A2: set_o] :
      ( ( ? [X3: $o] : ( member_o @ X3 @ A2 ) )
      = ( A2 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_287_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
      = ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_288_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ? [X3: int] : ( member_int @ X3 @ A2 ) )
      = ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_289_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y3: set_nat] :
          ~ ( member_set_nat @ Y3 @ A2 )
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_290_equals0I,axiom,
    ! [A2: set_set_nat_rat] :
      ( ! [Y3: set_nat_rat] :
          ~ ( member_set_nat_rat @ Y3 @ A2 )
     => ( A2 = bot_bo6797373522285170759at_rat ) ) ).

% equals0I
thf(fact_291_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y3: real] :
          ~ ( member_real @ Y3 @ A2 )
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_292_equals0I,axiom,
    ! [A2: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o @ Y3 @ A2 )
     => ( A2 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_293_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A2 )
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_294_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y3: int] :
          ~ ( member_int @ Y3 @ A2 )
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_295_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_296_equals0D,axiom,
    ! [A2: set_set_nat_rat,A: set_nat_rat] :
      ( ( A2 = bot_bo6797373522285170759at_rat )
     => ~ ( member_set_nat_rat @ A @ A2 ) ) ).

% equals0D
thf(fact_297_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ ( member_real @ A @ A2 ) ) ).

% equals0D
thf(fact_298_equals0D,axiom,
    ! [A2: set_o,A: $o] :
      ( ( A2 = bot_bot_set_o )
     => ~ ( member_o @ A @ A2 ) ) ).

% equals0D
thf(fact_299_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_300_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ ( member_int @ A @ A2 ) ) ).

% equals0D
thf(fact_301_emptyE,axiom,
    ! [A: set_nat] :
      ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_302_emptyE,axiom,
    ! [A: set_nat_rat] :
      ~ ( member_set_nat_rat @ A @ bot_bo6797373522285170759at_rat ) ).

% emptyE
thf(fact_303_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_304_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_305_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_306_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_307_psubsetD,axiom,
    ! [A2: set_o,B2: set_o,C: $o] :
      ( ( ord_less_set_o @ A2 @ B2 )
     => ( ( member_o @ C @ A2 )
       => ( member_o @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_308_psubsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B2 )
     => ( ( member_set_nat @ C @ A2 )
       => ( member_set_nat @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_309_psubsetD,axiom,
    ! [A2: set_set_nat_rat,B2: set_set_nat_rat,C: set_nat_rat] :
      ( ( ord_le1311537459589289991at_rat @ A2 @ B2 )
     => ( ( member_set_nat_rat @ C @ A2 )
       => ( member_set_nat_rat @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_310_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_311_psubsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_312_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_313_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_314_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_315_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_316_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_317_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_318_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_319_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_320_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_321_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_322_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_323_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_324_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_325_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_326_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_327_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_328_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_329_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_330_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_331_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_332_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_333_not__psubset__empty,axiom,
    ! [A2: set_real] :
      ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_334_not__psubset__empty,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_335_not__psubset__empty,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_336_not__psubset__empty,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_337_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_338_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_339_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_340_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_341_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_342_order__refl,axiom,
    ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).

% order_refl
thf(fact_343_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_344_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_345_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_346_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_347_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( ( member_complex @ X2 @ S2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_348_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ~ ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ S2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic1189837152898106425t_real @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_349_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( ( member_real @ X2 @ S2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_350_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_o,F: $o > real] :
      ( ( finite_finite_o @ S2 )
     => ( ( S2 != bot_bot_set_o )
       => ~ ? [X2: $o] :
              ( ( member_o @ X2 @ S2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8697145971487455083o_real @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_351_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ S2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_352_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( ( member_int @ X2 @ S2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_353_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( ( member_complex @ X2 @ S2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_354_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ~ ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ S2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic3210252021154270693at_rat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_355_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( ( member_real @ X2 @ S2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_356_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_o,F: $o > rat] :
      ( ( finite_finite_o @ S2 )
     => ( ( S2 != bot_bot_set_o )
       => ~ ? [X2: $o] :
              ( ( member_o @ X2 @ S2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_357_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( S2 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S2 )
         => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_358_arg__min__least,axiom,
    ! [S2: set_Extended_enat,Y: extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ( ( member_Extended_enat @ Y @ S2 )
         => ( ord_less_eq_rat @ ( F @ ( lattic3210252021154270693at_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_359_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ( ( member_real @ Y @ S2 )
         => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_360_arg__min__least,axiom,
    ! [S2: set_o,Y: $o,F: $o > rat] :
      ( ( finite_finite_o @ S2 )
     => ( ( S2 != bot_bot_set_o )
       => ( ( member_o @ Y @ S2 )
         => ( ord_less_eq_rat @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_361_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S2 )
         => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_362_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ( ( member_int @ Y @ S2 )
         => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_363_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( S2 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S2 )
         => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_364_arg__min__least,axiom,
    ! [S2: set_Extended_enat,Y: extended_enat,F: extended_enat > num] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ( ( member_Extended_enat @ Y @ S2 )
         => ( ord_less_eq_num @ ( F @ ( lattic402713867396545063at_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_365_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: real > num] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ( ( member_real @ Y @ S2 )
         => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_366_arg__min__least,axiom,
    ! [S2: set_o,Y: $o,F: $o > num] :
      ( ( finite_finite_o @ S2 )
     => ( ( S2 != bot_bot_set_o )
       => ( ( member_o @ Y @ S2 )
         => ( ord_less_eq_num @ ( F @ ( lattic8556559851467007577_o_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_367_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M2: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K2 @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K2 ) ) )
       => ( P @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_368_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
      = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).

% bot.not_eq_extremum
thf(fact_369_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).

% bot.not_eq_extremum
thf(fact_370_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_371_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_372_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_373_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_374_bot_Oextremum__strict,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_375_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_376_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_377_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_378_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_379_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_380_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_381_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_382_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_383_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_384_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_385_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_386_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_387_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_388_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_389_bot_Oextremum,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% bot.extremum
thf(fact_390_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_391_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_392_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_393_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D2 )
       => ? [E: real] :
            ( ( ord_less_real @ zero_zero_real @ E )
            & ( ord_less_real @ E @ D1 )
            & ( ord_less_real @ E @ D2 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_394_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D2 )
       => ? [E: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E )
            & ( ord_less_rat @ E @ D1 )
            & ( ord_less_rat @ E @ D2 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_395_subsetI,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( member_o @ X4 @ B2 ) )
     => ( ord_less_eq_set_o @ A2 @ B2 ) ) ).

% subsetI
thf(fact_396_subsetI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ A2 )
         => ( member_set_nat @ X4 @ B2 ) )
     => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_397_subsetI,axiom,
    ! [A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ! [X4: set_nat_rat] :
          ( ( member_set_nat_rat @ X4 @ A2 )
         => ( member_set_nat_rat @ X4 @ B2 ) )
     => ( ord_le4375437777232675859at_rat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_398_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( member_nat @ X4 @ B2 ) )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_399_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( member_int @ X4 @ B2 ) )
     => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% subsetI
thf(fact_400_subset__antisym,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% subset_antisym
thf(fact_401_in__mono,axiom,
    ! [A2: set_o,B2: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A2 @ B2 )
     => ( ( member_o @ X @ A2 )
       => ( member_o @ X @ B2 ) ) ) ).

% in_mono
thf(fact_402_in__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( member_set_nat @ X @ A2 )
       => ( member_set_nat @ X @ B2 ) ) ) ).

% in_mono
thf(fact_403_in__mono,axiom,
    ! [A2: set_set_nat_rat,B2: set_set_nat_rat,X: set_nat_rat] :
      ( ( ord_le4375437777232675859at_rat @ A2 @ B2 )
     => ( ( member_set_nat_rat @ X @ A2 )
       => ( member_set_nat_rat @ X @ B2 ) ) ) ).

% in_mono
thf(fact_404_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( member_nat @ X @ A2 )
       => ( member_nat @ X @ B2 ) ) ) ).

% in_mono
thf(fact_405_in__mono,axiom,
    ! [A2: set_int,B2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( member_int @ X @ A2 )
       => ( member_int @ X @ B2 ) ) ) ).

% in_mono
thf(fact_406_subsetD,axiom,
    ! [A2: set_o,B2: set_o,C: $o] :
      ( ( ord_less_eq_set_o @ A2 @ B2 )
     => ( ( member_o @ C @ A2 )
       => ( member_o @ C @ B2 ) ) ) ).

% subsetD
thf(fact_407_subsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( member_set_nat @ C @ A2 )
       => ( member_set_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_408_subsetD,axiom,
    ! [A2: set_set_nat_rat,B2: set_set_nat_rat,C: set_nat_rat] :
      ( ( ord_le4375437777232675859at_rat @ A2 @ B2 )
     => ( ( member_set_nat_rat @ C @ A2 )
       => ( member_set_nat_rat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_409_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_410_subsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B2 ) ) ) ).

% subsetD
thf(fact_411_equalityE,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
         => ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).

% equalityE
thf(fact_412_subset__eq,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A6: set_o,B6: set_o] :
        ! [X3: $o] :
          ( ( member_o @ X3 @ A6 )
         => ( member_o @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_413_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A6 )
         => ( member_set_nat @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_414_subset__eq,axiom,
    ( ord_le4375437777232675859at_rat
    = ( ^ [A6: set_set_nat_rat,B6: set_set_nat_rat] :
        ! [X3: set_nat_rat] :
          ( ( member_set_nat_rat @ X3 @ A6 )
         => ( member_set_nat_rat @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_415_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ A6 )
         => ( member_nat @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_416_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [X3: int] :
          ( ( member_int @ X3 @ A6 )
         => ( member_int @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_417_equalityD1,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% equalityD1
thf(fact_418_equalityD2,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_int @ B2 @ A2 ) ) ).

% equalityD2
thf(fact_419_subset__iff,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A6: set_o,B6: set_o] :
        ! [T2: $o] :
          ( ( member_o @ T2 @ A6 )
         => ( member_o @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_420_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [T2: set_nat] :
          ( ( member_set_nat @ T2 @ A6 )
         => ( member_set_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_421_subset__iff,axiom,
    ( ord_le4375437777232675859at_rat
    = ( ^ [A6: set_set_nat_rat,B6: set_set_nat_rat] :
        ! [T2: set_nat_rat] :
          ( ( member_set_nat_rat @ T2 @ A6 )
         => ( member_set_nat_rat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_422_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A6 )
         => ( member_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_423_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A6 )
         => ( member_int @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_424_subset__refl,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).

% subset_refl
thf(fact_425_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X4: set_nat] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_426_Collect__mono,axiom,
    ! [P: set_nat_rat > $o,Q: set_nat_rat > $o] :
      ( ! [X4: set_nat_rat] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le4375437777232675859at_rat @ ( collect_set_nat_rat @ P ) @ ( collect_set_nat_rat @ Q ) ) ) ).

% Collect_mono
thf(fact_427_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_428_Collect__mono,axiom,
    ! [P: ( nat > rat ) > $o,Q: ( nat > rat ) > $o] :
      ( ! [X4: nat > rat] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le2679597024174929757at_rat @ ( collect_nat_rat @ P ) @ ( collect_nat_rat @ Q ) ) ) ).

% Collect_mono
thf(fact_429_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_430_subset__trans,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C2 )
       => ( ord_less_eq_set_int @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_431_set__eq__subset,axiom,
    ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_432_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X3: set_nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_433_Collect__mono__iff,axiom,
    ! [P: set_nat_rat > $o,Q: set_nat_rat > $o] :
      ( ( ord_le4375437777232675859at_rat @ ( collect_set_nat_rat @ P ) @ ( collect_set_nat_rat @ Q ) )
      = ( ! [X3: set_nat_rat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_434_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_435_Collect__mono__iff,axiom,
    ! [P: ( nat > rat ) > $o,Q: ( nat > rat ) > $o] :
      ( ( ord_le2679597024174929757at_rat @ ( collect_nat_rat @ P ) @ ( collect_nat_rat @ Q ) )
      = ( ! [X3: nat > rat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_436_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X3: int] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_437_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_438_bot__set__def,axiom,
    ( bot_bo6797373522285170759at_rat
    = ( collect_set_nat_rat @ bot_bo3445895781125589758_rat_o ) ) ).

% bot_set_def
thf(fact_439_bot__set__def,axiom,
    ( bot_bot_set_nat_rat
    = ( collect_nat_rat @ bot_bot_nat_rat_o ) ) ).

% bot_set_def
thf(fact_440_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_441_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_442_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_443_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_444_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_445_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_446_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_447_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_448_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_449_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y @ X )
         => ~ ( ord_less_eq_rat @ X @ Z ) )
       => ( ( ( ord_less_eq_rat @ X @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X )
                 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_450_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ( ord_less_eq_num @ X @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z ) )
     => ( ( ( ord_less_eq_num @ Y @ X )
         => ~ ( ord_less_eq_num @ X @ Z ) )
       => ( ( ( ord_less_eq_num @ X @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y ) )
         => ( ( ( ord_less_eq_num @ Z @ Y )
             => ~ ( ord_less_eq_num @ Y @ X ) )
           => ( ( ( ord_less_eq_num @ Y @ Z )
               => ~ ( ord_less_eq_num @ Z @ X ) )
             => ~ ( ( ord_less_eq_num @ Z @ X )
                 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_451_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_452_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z ) )
       => ( ( ( ord_less_eq_int @ X @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X ) )
             => ~ ( ( ord_less_eq_int @ Z @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_453_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
    = ( ^ [X3: set_int,Y2: set_int] :
          ( ( ord_less_eq_set_int @ X3 @ Y2 )
          & ( ord_less_eq_set_int @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_454_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y2 )
          & ( ord_less_eq_rat @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_455_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_eq_num @ X3 @ Y2 )
          & ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_456_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_457_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_458_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_459_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_460_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_461_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_462_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_463_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_464_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_465_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_466_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_467_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_468_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_469_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_470_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_471_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_472_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_473_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% order.trans
thf(fact_474_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_475_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_476_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_477_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_478_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_eq_set_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_479_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_eq_rat @ X @ Z ) ) ) ).

% order_trans
thf(fact_480_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_eq_num @ X @ Z ) ) ) ).

% order_trans
thf(fact_481_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_482_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_483_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A5: rat,B5: rat] :
          ( ( ord_less_eq_rat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: rat,B5: rat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_484_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A5: num,B5: num] :
          ( ( ord_less_eq_num @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: num,B5: num] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_485_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat,B5: nat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_486_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: int,B5: int] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_487_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A4 )
          & ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_488_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_489_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_490_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_491_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_492_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_493_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_494_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_495_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_496_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_497_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_eq_set_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_498_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_499_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_500_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_501_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_502_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_503_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_504_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_505_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_506_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_507_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A4 @ B4 )
          & ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_508_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_509_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_510_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_511_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_512_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_513_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_514_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 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_515_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 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_eq_int @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_516_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_517_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_518_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 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_519_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 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_eq_int @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_520_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_521_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_522_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_523_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_524_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_525_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_526_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_527_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_528_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_529_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_530_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 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_531_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 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_532_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_533_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_534_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_eq_refl
thf(fact_535_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_536_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_537_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_linear
thf(fact_538_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_linear
thf(fact_539_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_540_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_541_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_542_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_543_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_544_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_545_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_546_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_547_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_548_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_549_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_550_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_551_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_552_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_553_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_554_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_555_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_556_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_557_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_558_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_559_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_560_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_561_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_562_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_563_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_564_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_565_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_less_eq_set_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_566_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_567_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_568_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_569_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_570_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).

% lt_ex
thf(fact_571_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).

% lt_ex
thf(fact_572_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_573_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).

% gt_ex
thf(fact_574_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).

% gt_ex
thf(fact_575_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_576_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_577_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z3: real] :
          ( ( ord_less_real @ X @ Z3 )
          & ( ord_less_real @ Z3 @ Y ) ) ) ).

% dense
thf(fact_578_dense,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ? [Z3: rat] :
          ( ( ord_less_rat @ X @ Z3 )
          & ( ord_less_rat @ Z3 @ Y ) ) ) ).

% dense
thf(fact_579_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_580_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_581_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_582_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_583_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_584_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_585_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_586_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_587_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_588_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_589_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_590_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_591_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_592_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_593_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_594_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_595_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_596_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_597_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_598_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_599_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y4: nat] :
              ( ( ord_less_nat @ Y4 @ X4 )
             => ( P @ Y4 ) )
         => ( P @ X4 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_600_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_601_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_602_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_603_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_604_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_605_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_606_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_607_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_608_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_609_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_610_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_611_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_612_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_613_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_614_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_615_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_616_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_617_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_618_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_619_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_620_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X6: nat] : ( P2 @ X6 ) )
    = ( ^ [P3: nat > $o] :
        ? [N4: nat] :
          ( ( P3 @ N4 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
             => ~ ( P3 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_621_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B: real] :
      ( ! [A5: real,B5: real] :
          ( ( ord_less_real @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: real] : ( P @ A5 @ A5 )
       => ( ! [A5: real,B5: real] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_622_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A5: rat,B5: rat] :
          ( ( ord_less_rat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: rat] : ( P @ A5 @ A5 )
       => ( ! [A5: rat,B5: rat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_623_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A5: num,B5: num] :
          ( ( ord_less_num @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: num] : ( P @ A5 @ A5 )
       => ( ! [A5: num,B5: num] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_624_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B5: nat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_625_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_int @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: int] : ( P @ A5 @ A5 )
       => ( ! [A5: int,B5: int] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_626_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_627_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_628_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_629_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_630_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_631_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ( ord_less_real @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_632_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_633_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ( ord_less_num @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_634_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_635_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_636_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_637_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_638_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_639_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_640_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_641_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_642_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_643_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_644_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_645_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_646_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_647_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_648_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_649_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_650_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_651_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_652_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_653_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_654_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_655_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_656_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_657_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_658_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_659_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_660_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_661_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      = ( ( ord_less_real @ X @ Y )
        | ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_662_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      = ( ( ord_less_rat @ X @ Y )
        | ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_663_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      = ( ( ord_less_num @ X @ Y )
        | ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_664_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_665_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_666_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_667_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_668_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_669_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_670_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_671_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_672_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_673_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_674_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_675_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_676_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_677_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_678_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_679_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_680_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_681_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_682_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_683_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_684_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_685_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_686_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_687_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_688_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_689_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_690_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_691_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_692_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_693_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_694_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_695_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_696_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_697_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_698_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_699_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_700_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_701_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_702_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_703_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_704_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_705_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_706_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_707_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_708_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_709_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_710_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_711_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_712_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_713_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_714_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_715_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_716_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_717_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_718_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_719_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_720_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_721_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_722_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_723_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_724_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_725_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_726_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: $o] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_727_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: $o] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_728_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: $o] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_729_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_730_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_731_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
      | ( X = Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_732_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
      | ( X = Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_733_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
      | ( X = Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_734_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_735_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_736_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_737_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_738_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_739_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_740_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_741_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_742_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_743_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_744_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_745_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_746_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_747_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_748_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_749_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_750_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_751_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_752_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ~ ( ord_less_set_int @ X @ Y ) ) ).

% leD
thf(fact_753_leD,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ~ ( ord_less_rat @ X @ Y ) ) ).

% leD
thf(fact_754_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_755_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_756_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_757_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_758_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% leI
thf(fact_759_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_760_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_761_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_762_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_763_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ~ ( ord_less_set_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_764_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_765_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_766_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_767_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_768_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_769_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ ( ord_less_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_770_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_771_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_772_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_773_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_774_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_775_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ~ ( ord_less_set_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_776_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_777_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_778_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_779_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_780_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X4: real] :
          ( ( ord_less_real @ Z @ X4 )
         => ( ord_less_eq_real @ Y @ X4 ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_ge
thf(fact_781_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_rat @ Z @ X4 )
         => ( ord_less_eq_rat @ Y @ X4 ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_ge
thf(fact_782_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X4: real] :
          ( ( ord_less_real @ X4 @ Y )
         => ( ord_less_eq_real @ X4 @ Z ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_le
thf(fact_783_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Y )
         => ( ord_less_eq_rat @ X4 @ Z ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_le
thf(fact_784_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_eq_real @ X3 @ Y2 )
          & ~ ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_785_less__le__not__le,axiom,
    ( ord_less_set_int
    = ( ^ [X3: set_int,Y2: set_int] :
          ( ( ord_less_eq_set_int @ X3 @ Y2 )
          & ~ ( ord_less_eq_set_int @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_786_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y2 )
          & ~ ( ord_less_eq_rat @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_787_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_eq_num @ X3 @ Y2 )
          & ~ ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_788_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_789_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ~ ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_790_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_eq_real @ Y @ X )
     => ( ord_less_real @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_791_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_rat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_792_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_eq_num @ Y @ X )
     => ( ord_less_num @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_793_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_794_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_795_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_real @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_796_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_set_int @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_797_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_rat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_798_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_num @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_799_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_800_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_int @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_801_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_802_order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_803_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_804_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_805_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_806_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_807_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_808_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_set_int @ B @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_809_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_810_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_811_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_812_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_813_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_814_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_815_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_816_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_817_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_818_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_819_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
          & ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_820_order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A4 @ B4 )
          & ~ ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_821_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ~ ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_822_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ~ ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_823_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_824_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_825_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ X )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z @ W )
           => ( ( ord_less_real @ W @ X )
             => ( ord_less_eq_real @ Y @ W ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_826_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ X )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ Z @ W )
           => ( ( ord_less_rat @ W @ X )
             => ( ord_less_eq_rat @ Y @ W ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_827_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ! [W: real] :
            ( ( ord_less_real @ X @ W )
           => ( ( ord_less_real @ W @ Y )
             => ( ord_less_eq_real @ W @ Z ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_828_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ X @ W )
           => ( ( ord_less_rat @ W @ Y )
             => ( ord_less_eq_rat @ W @ Z ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_829_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_real @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_830_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( ( ord_less_set_int @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_831_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_rat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_832_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_num @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_833_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_834_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_int @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_835_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_eq_real @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_836_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_837_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_838_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_839_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_840_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_841_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_842_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_set_int @ C @ B )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_843_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_844_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_845_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_846_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_847_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_848_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_849_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_850_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_851_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_852_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_853_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_eq_real @ B4 @ A4 )
          & ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_854_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A4 )
          & ~ ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_855_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ~ ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_856_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ~ ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_857_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_858_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_859_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_860_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ord_less_eq_set_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_861_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_862_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_863_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_864_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_865_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_eq_real @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_866_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ord_less_eq_set_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_867_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_868_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_869_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_870_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_871_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_real @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_872_order__le__less,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X3: set_int,Y2: set_int] :
          ( ( ord_less_set_int @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_873_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_rat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_874_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_num @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_875_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_nat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_876_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_int @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_877_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_eq_real @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_878_order__less__le,axiom,
    ( ord_less_set_int
    = ( ^ [X3: set_int,Y2: set_int] :
          ( ( ord_less_eq_set_int @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_879_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_880_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_eq_num @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_881_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_882_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_883_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_884_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
      = ( ord_less_rat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_885_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_886_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_887_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_888_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_889_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_890_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_891_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_892_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_893_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_894_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_set_int @ X @ Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_895_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_896_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_897_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_898_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_899_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( A != B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_900_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_901_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_902_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_903_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_904_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_905_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_906_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( ord_less_set_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_907_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_908_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_909_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_910_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_911_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_912_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_set_int @ Y @ Z )
       => ( ord_less_set_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_913_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_914_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_915_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_916_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_917_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_918_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_set_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_919_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_920_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_921_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_922_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_923_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_924_order__le__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_925_order__le__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_926_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_927_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_928_order__le__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_929_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_930_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_931_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 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_932_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 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_933_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_934_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_935_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_936_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_937_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_938_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_939_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_940_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_941_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_942_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_943_order__less__le__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_944_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_945_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_946_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_947_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_948_order__less__le__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_949_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_950_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_951_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_952_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_953_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_954_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_955_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_956_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_957_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_958_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_959_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 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_960_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 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_961_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 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_962_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 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_963_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_964_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_965_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_966_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_967_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_968_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_969_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_set_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_970_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_971_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_972_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_973_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_974_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X4: set_nat] :
          ~ ( member_set_nat @ X4 @ A2 )
     => ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_975_subset__emptyI,axiom,
    ! [A2: set_set_nat_rat] :
      ( ! [X4: set_nat_rat] :
          ~ ( member_set_nat_rat @ X4 @ A2 )
     => ( ord_le4375437777232675859at_rat @ A2 @ bot_bo6797373522285170759at_rat ) ) ).

% subset_emptyI
thf(fact_976_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X4: real] :
          ~ ( member_real @ X4 @ A2 )
     => ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_977_subset__emptyI,axiom,
    ! [A2: set_o] :
      ( ! [X4: $o] :
          ~ ( member_o @ X4 @ A2 )
     => ( ord_less_eq_set_o @ A2 @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_978_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] :
          ~ ( member_nat @ X4 @ A2 )
     => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_979_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] :
          ~ ( member_int @ X4 @ A2 )
     => ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_980_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ~ ( ord_less_eq_real @ T @ X2 ) ) ).

% minf(8)
thf(fact_981_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ~ ( ord_less_eq_rat @ T @ X2 ) ) ).

% minf(8)
thf(fact_982_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ~ ( ord_less_eq_num @ T @ X2 ) ) ).

% minf(8)
thf(fact_983_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).

% minf(8)
thf(fact_984_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ~ ( ord_less_eq_int @ T @ X2 ) ) ).

% minf(8)
thf(fact_985_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ( ord_less_eq_real @ X2 @ T ) ) ).

% minf(6)
thf(fact_986_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ( ord_less_eq_rat @ X2 @ T ) ) ).

% minf(6)
thf(fact_987_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ( ord_less_eq_num @ X2 @ T ) ) ).

% minf(6)
thf(fact_988_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ( ord_less_eq_nat @ X2 @ T ) ) ).

% minf(6)
thf(fact_989_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ( ord_less_eq_int @ X2 @ T ) ) ).

% minf(6)
thf(fact_990_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ( ord_less_eq_real @ T @ X2 ) ) ).

% pinf(8)
thf(fact_991_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ( ord_less_eq_rat @ T @ X2 ) ) ).

% pinf(8)
thf(fact_992_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ( ord_less_eq_num @ T @ X2 ) ) ).

% pinf(8)
thf(fact_993_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ( ord_less_eq_nat @ T @ X2 ) ) ).

% pinf(8)
thf(fact_994_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ( ord_less_eq_int @ T @ X2 ) ) ).

% pinf(8)
thf(fact_995_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ~ ( ord_less_eq_real @ X2 @ T ) ) ).

% pinf(6)
thf(fact_996_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ~ ( ord_less_eq_rat @ X2 @ T ) ) ).

% pinf(6)
thf(fact_997_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ~ ( ord_less_eq_num @ X2 @ T ) ) ).

% pinf(6)
thf(fact_998_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).

% pinf(6)
thf(fact_999_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ~ ( ord_less_eq_int @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1000_verit__comp__simplify1_I3_J,axiom,
    ! [B7: real,A7: real] :
      ( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
      = ( ord_less_real @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1001_verit__comp__simplify1_I3_J,axiom,
    ! [B7: rat,A7: rat] :
      ( ( ~ ( ord_less_eq_rat @ B7 @ A7 ) )
      = ( ord_less_rat @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1002_verit__comp__simplify1_I3_J,axiom,
    ! [B7: num,A7: num] :
      ( ( ~ ( ord_less_eq_num @ B7 @ A7 ) )
      = ( ord_less_num @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1003_verit__comp__simplify1_I3_J,axiom,
    ! [B7: nat,A7: nat] :
      ( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
      = ( ord_less_nat @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1004_verit__comp__simplify1_I3_J,axiom,
    ! [B7: int,A7: int] :
      ( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
      = ( ord_less_int @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1005_complete__interval,axiom,
    ! [A: real,B: real,P: real > $o] :
      ( ( ord_less_real @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: real] :
              ( ( ord_less_eq_real @ A @ C3 )
              & ( ord_less_eq_real @ C3 @ B )
              & ! [X2: real] :
                  ( ( ( ord_less_eq_real @ A @ X2 )
                    & ( ord_less_real @ X2 @ C3 ) )
                 => ( P @ X2 ) )
              & ! [D3: real] :
                  ( ! [X4: real] :
                      ( ( ( ord_less_eq_real @ A @ X4 )
                        & ( ord_less_real @ X4 @ D3 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1006_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A @ C3 )
              & ( ord_less_eq_nat @ C3 @ B )
              & ! [X2: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X2 )
                    & ( ord_less_nat @ X2 @ C3 ) )
                 => ( P @ X2 ) )
              & ! [D3: nat] :
                  ( ! [X4: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X4 )
                        & ( ord_less_nat @ X4 @ D3 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1007_complete__interval,axiom,
    ! [A: int,B: int,P: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: int] :
              ( ( ord_less_eq_int @ A @ C3 )
              & ( ord_less_eq_int @ C3 @ B )
              & ! [X2: int] :
                  ( ( ( ord_less_eq_int @ A @ X2 )
                    & ( ord_less_int @ X2 @ C3 ) )
                 => ( P @ X2 ) )
              & ! [D3: int] :
                  ( ! [X4: int] :
                      ( ( ( ord_less_eq_int @ A @ X4 )
                        & ( ord_less_int @ X4 @ D3 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1008_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S3 ) ) ) ).

% deg_SUcn_Node
thf(fact_1009_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_1010_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_1011_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_1012_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_1013_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] :
      ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
      = zero_zero_nat ) ).

% VEBT.size_gen(2)
thf(fact_1014_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_1015_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).

% not_min_Null_member
thf(fact_1016_nat_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( ( suc @ X23 )
        = ( suc @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% nat.inject
thf(fact_1017_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1018_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_eq
thf(fact_1019_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1020_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1021_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% Suc_le_mono
thf(fact_1022_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1023_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1024_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_1025_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1026_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_1027_Zero__not__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_not_Suc
thf(fact_1028_Zero__neq__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_neq_Suc
thf(fact_1029_Suc__neq__Zero,axiom,
    ! [M2: nat] :
      ( ( suc @ M2 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1030_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1031_diff__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N: nat] :
      ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X4: nat,Y3: nat] :
              ( ( P @ X4 @ Y3 )
             => ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
         => ( P @ M2 @ N ) ) ) ) ).

% diff_induct
thf(fact_1032_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_1033_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va: nat] :
              ( X
             != ( suc @ ( suc @ Va ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_1034_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1035_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat
        = ( suc @ X23 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1036_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1037_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1038_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] :
      ( zero_zero_nat
     != ( suc @ X23 ) ) ).

% nat.distinct(1)
thf(fact_1039_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1040_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_1041_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I2 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1042_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1043_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_SucD
thf(fact_1044_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
       => ( M2 = N ) ) ) ).

% less_antisym
thf(fact_1045_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M2 )
      = ( ? [M6: nat] :
            ( ( M2
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1046_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_1047_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_1048_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) ) ) ).

% less_Suc_eq
thf(fact_1049_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_1050_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1051_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( M2 = N ) ) ) ).

% less_SucE
thf(fact_1052_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ( suc @ M2 )
         != N )
       => ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1053_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_1054_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_lessD
thf(fact_1055_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_1056_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_leD
thf(fact_1057_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( M2
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1058_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1059_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
     => ? [M4: nat] :
          ( M7
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_1060_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M2 @ N )
        | ( M2
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1061_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1062_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).

% not_less_eq_eq
thf(fact_1063_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
             => ( P @ M ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1064_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( P @ M2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1065_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ! [X4: nat] : ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M2 @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1066_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_1067_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_1068_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_1069_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1070_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1071_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1072_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1073_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1074_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1075_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1076_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1077_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1078_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1079_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1080_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1081_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1082_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1083_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1084_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_set_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1085_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_rat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1086_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_num @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1087_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1088_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1089_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M2
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1090_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_1091_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_1092_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1093_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_1094_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_leI
thf(fact_1095_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_eq
thf(fact_1096_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1097_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_1098_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_lessD
thf(fact_1099_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_1100_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_Suc_eq_le
thf(fact_1101_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1102_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1103_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1104_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
      = ( vEBT_Leaf @ A @ B ) ) ).

% vebt_delete.simps(3)
thf(fact_1105_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I3: nat] :
                ( ( ord_less_eq_nat @ I3 @ K2 )
               => ~ ( P @ I3 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1106_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_1107_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_1108_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_1109_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_1110_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1111_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1112_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1113_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1114_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1115_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B5: real] :
      ( ( ord_less_real @ A @ B5 )
      | ( ord_less_real @ B5 @ A ) ) ).

% ex_gt_or_lt
thf(fact_1116_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1117_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1118_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1119_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1120_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1121_pinf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X4: real] :
          ( ( ord_less_real @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( ( ord_less_real @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1122_pinf_I1_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1123_pinf_I1_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X4: num] :
          ( ( ord_less_num @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( ( ord_less_num @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1124_pinf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1125_pinf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X4: int] :
          ( ( ord_less_int @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1126_pinf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X4: real] :
          ( ( ord_less_real @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( ( ord_less_real @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1127_pinf_I2_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1128_pinf_I2_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X4: num] :
          ( ( ord_less_num @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( ( ord_less_num @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1129_pinf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1130_pinf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X4: int] :
          ( ( ord_less_int @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z5 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1131_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1132_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1133_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1134_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1135_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1136_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1137_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1138_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1139_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1140_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1141_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ~ ( ord_less_real @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1142_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ~ ( ord_less_rat @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1143_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ~ ( ord_less_num @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1144_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ~ ( ord_less_nat @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1145_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ~ ( ord_less_int @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1146_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ( ord_less_real @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1147_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ( ord_less_rat @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1148_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ( ord_less_num @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1149_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ( ord_less_nat @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1150_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ( ord_less_int @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1151_minf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X4: real] :
          ( ( ord_less_real @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( ( ord_less_real @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1152_minf_I1_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1153_minf_I1_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X4: num] :
          ( ( ord_less_num @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( ( ord_less_num @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1154_minf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1155_minf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1156_minf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X4: real] :
          ( ( ord_less_real @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( ( ord_less_real @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1157_minf_I2_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1158_minf_I2_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X4: num] :
          ( ( ord_less_num @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( ( ord_less_num @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1159_minf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1160_minf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q2 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1161_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1162_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1163_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1164_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1165_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1166_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1167_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1168_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1169_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1170_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1171_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ( ord_less_real @ X2 @ T ) ) ).

% minf(5)
thf(fact_1172_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ( ord_less_rat @ X2 @ T ) ) ).

% minf(5)
thf(fact_1173_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ( ord_less_num @ X2 @ T ) ) ).

% minf(5)
thf(fact_1174_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ( ord_less_nat @ X2 @ T ) ) ).

% minf(5)
thf(fact_1175_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ( ord_less_int @ X2 @ T ) ) ).

% minf(5)
thf(fact_1176_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ~ ( ord_less_real @ T @ X2 ) ) ).

% minf(7)
thf(fact_1177_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ~ ( ord_less_rat @ T @ X2 ) ) ).

% minf(7)
thf(fact_1178_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ~ ( ord_less_num @ T @ X2 ) ) ).

% minf(7)
thf(fact_1179_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ~ ( ord_less_nat @ T @ X2 ) ) ).

% minf(7)
thf(fact_1180_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ~ ( ord_less_int @ T @ X2 ) ) ).

% minf(7)
thf(fact_1181_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1182_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_1183_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_1184_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_1185_Collect__empty__eq__bot,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( P = bot_bot_set_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1186_Collect__empty__eq__bot,axiom,
    ! [P: set_nat_rat > $o] :
      ( ( ( collect_set_nat_rat @ P )
        = bot_bo6797373522285170759at_rat )
      = ( P = bot_bo3445895781125589758_rat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1187_Collect__empty__eq__bot,axiom,
    ! [P: ( nat > rat ) > $o] :
      ( ( ( collect_nat_rat @ P )
        = bot_bot_set_nat_rat )
      = ( P = bot_bot_nat_rat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1188_Collect__empty__eq__bot,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( P = bot_bot_real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1189_Collect__empty__eq__bot,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( P = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1190_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1191_Collect__empty__eq__bot,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( P = bot_bot_int_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1192_bot__empty__eq,axiom,
    ( bot_bot_set_nat_o
    = ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1193_bot__empty__eq,axiom,
    ( bot_bo3445895781125589758_rat_o
    = ( ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ bot_bo6797373522285170759at_rat ) ) ) ).

% bot_empty_eq
thf(fact_1194_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X3: real] : ( member_real @ X3 @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_1195_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X3: $o] : ( member_o @ X3 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_1196_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1197_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X3: int] : ( member_int @ X3 @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_1198_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N2: nat] :
            ( ~ ( P @ N2 )
            & ( P @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1199_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_1200_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_1201_Set_Ois__empty__def,axiom,
    ( is_empty_real
    = ( ^ [A6: set_real] : ( A6 = bot_bot_set_real ) ) ) ).

% Set.is_empty_def
thf(fact_1202_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A6: set_o] : ( A6 = bot_bot_set_o ) ) ) ).

% Set.is_empty_def
thf(fact_1203_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A6: set_nat] : ( A6 = bot_bot_set_nat ) ) ) ).

% Set.is_empty_def
thf(fact_1204_Set_Ois__empty__def,axiom,
    ( is_empty_int
    = ( ^ [A6: set_int] : ( A6 = bot_bot_set_int ) ) ) ).

% Set.is_empty_def
thf(fact_1205_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_1206_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_1207_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_1208_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_1209_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_1210_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
      = zero_zero_nat ) ).

% VEBT.size(4)
thf(fact_1211_enumerate__mono__iff,axiom,
    ! [S2: set_Extended_enat,M2: nat,N: nat] :
      ( ~ ( finite4001608067531595151d_enat @ S2 )
     => ( ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ M2 ) @ ( infini7641415182203889163d_enat @ S2 @ N ) )
        = ( ord_less_nat @ M2 @ N ) ) ) ).

% enumerate_mono_iff
thf(fact_1212_enumerate__mono__iff,axiom,
    ! [S2: set_nat,M2: nat,N: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) )
        = ( ord_less_nat @ M2 @ N ) ) ) ).

% enumerate_mono_iff
thf(fact_1213_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_1214_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M2 )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_1215_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ M2 )
        = ( semiri681578069525770553at_rat @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_1216_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_1217_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1218_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1219_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_1220_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1221_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1222_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1223_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_1224_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1225_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1226_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1227_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1228_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_1229_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_1230_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_1231_Suc__diff__diff,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1232_diff__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_Suc_Suc
thf(fact_1233_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1234_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1235_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1236_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1237_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_1238_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_1239_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1240_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_1241_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_1242_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_1243_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1244_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1245_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1246_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_1247_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_1248_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_1249_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_1250_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_1251_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_1252_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_1253_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_1254_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M2 )
        = zero_zero_nat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_1255_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = zero_zero_int )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_1256_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M2 )
        = zero_zero_real )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_1257_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri681578069525770553at_rat @ M2 )
        = zero_zero_rat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_1258_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_1259_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_1260_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_1261_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_1262_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_1263_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_1264_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_1265_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_1266_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% zero_less_diff
thf(fact_1267_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1268_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% diff_is_0_eq
thf(fact_1269_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_1270_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_1271_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_1272_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri5074537144036343181t_real @ N )
        = one_one_real )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_1273_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_1274_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_1275_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_1276_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_1277_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_1278_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_1279_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_1280_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_1281_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_1282_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_1283_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_1284_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1285_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_1286_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_1287_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_1288_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_1289_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_1290_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_1291_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_1292_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% of_nat_diff
thf(fact_1293_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_1294_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_1295_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1296_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_1297_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_1298_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_1299_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1300_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_1301_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_1302_real__arch__simple,axiom,
    ! [X: real] :
    ? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% real_arch_simple
thf(fact_1303_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% real_arch_simple
thf(fact_1304_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% reals_Archimedean2
thf(fact_1305_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% reals_Archimedean2
thf(fact_1306_size__neq__size__imp__neq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_size_VEBT_VEBT @ X )
       != ( size_size_VEBT_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1307_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X )
       != ( size_s6755466524823107622T_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1308_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ( size_size_num @ X )
       != ( size_size_num @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1309_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1310_size__neq__size__imp__neq,axiom,
    ! [X: char,Y: char] :
      ( ( ( size_size_char @ X )
       != ( size_size_char @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1311_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1312_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_1313_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_1314_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1315_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_1316_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_1317_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1318_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_1319_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_1320_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1321_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_1322_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_1323_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
    = ( ^ [A4: real,B4: real] :
          ( ( minus_minus_real @ A4 @ B4 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1324_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
    = ( ^ [A4: rat,B4: rat] :
          ( ( minus_minus_rat @ A4 @ B4 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1325_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
    = ( ^ [A4: int,B4: int] :
          ( ( minus_minus_int @ A4 @ B4 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1326_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1327_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_1328_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_1329_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1330_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_1331_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_1332_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1333_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_1334_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_1335_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1336_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_1337_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_1338_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_1339_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_1340_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_1341_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_1342_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_1343_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M2 )
          = zero_zero_nat )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
thf(fact_1344_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_1345_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_1346_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_1347_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_1348_diff__le__self,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).

% diff_le_self
thf(fact_1349_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1350_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1351_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1352_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M2 = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1353_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_1354_enumerate__in__set,axiom,
    ! [S2: set_Extended_enat,N: nat] :
      ( ~ ( finite4001608067531595151d_enat @ S2 )
     => ( member_Extended_enat @ ( infini7641415182203889163d_enat @ S2 @ N ) @ S2 ) ) ).

% enumerate_in_set
thf(fact_1355_enumerate__in__set,axiom,
    ! [S2: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( member_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ S2 ) ) ).

% enumerate_in_set
thf(fact_1356_enumerate__Ex,axiom,
    ! [S2: set_nat,S: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( ( member_nat @ S @ S2 )
       => ? [N2: nat] :
            ( ( infini8530281810654367211te_nat @ S2 @ N2 )
            = S ) ) ) ).

% enumerate_Ex
thf(fact_1357_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).

% of_nat_0_le_iff
thf(fact_1358_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_1359_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_1360_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_1361_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_1362_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_1363_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_1364_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_1365_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_1366_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_1367_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_1368_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_1369_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_1370_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_1371_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_1372_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_1373_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_1374_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_1375_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_1376_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_1377_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).

% of_nat_mono
thf(fact_1378_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_1379_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_1380_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_1381_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_1382_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_1383_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_1384_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_1385_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_1386_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_1387_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_1388_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M2 @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1389_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).

% diff_less
thf(fact_1390_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1391_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_1392_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M2 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1393_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1394_le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).

% le_enumerate
thf(fact_1395_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1396_enumerate__step,axiom,
    ! [S2: set_Extended_enat,N: nat] :
      ( ~ ( finite4001608067531595151d_enat @ S2 )
     => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ N ) @ ( infini7641415182203889163d_enat @ S2 @ ( suc @ N ) ) ) ) ).

% enumerate_step
thf(fact_1397_enumerate__step,axiom,
    ! [S2: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ ( infini8530281810654367211te_nat @ S2 @ ( suc @ N ) ) ) ) ).

% enumerate_step
thf(fact_1398_enumerate__mono,axiom,
    ! [M2: nat,N: nat,S2: set_Extended_enat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ~ ( finite4001608067531595151d_enat @ S2 )
       => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ M2 ) @ ( infini7641415182203889163d_enat @ S2 @ N ) ) ) ) ).

% enumerate_mono
thf(fact_1399_enumerate__mono,axiom,
    ! [M2: nat,N: nat,S2: set_nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ~ ( finite_finite_nat @ S2 )
       => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).

% enumerate_mono
thf(fact_1400_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_1401_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1402_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_1403_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_1404_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ( minus_minus_set_real @ X @ Y )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_1405_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( minus_minus_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_1406_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_1407_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( minus_minus_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_1408_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_1409_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% pos_int_cases
thf(fact_1410_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( K
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_1411_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_1412_frac__eq,axiom,
    ! [X: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = X )
      = ( ( ord_less_eq_real @ zero_zero_real @ X )
        & ( ord_less_real @ X @ one_one_real ) ) ) ).

% frac_eq
thf(fact_1413_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_1414_finite__enum__subset,axiom,
    ! [X5: set_Extended_enat,Y6: set_Extended_enat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( finite121521170596916366d_enat @ X5 ) )
         => ( ( infini7641415182203889163d_enat @ X5 @ I2 )
            = ( infini7641415182203889163d_enat @ Y6 @ I2 ) ) )
     => ( ( finite4001608067531595151d_enat @ X5 )
       => ( ( finite4001608067531595151d_enat @ Y6 )
         => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ X5 ) @ ( finite121521170596916366d_enat @ Y6 ) )
           => ( ord_le7203529160286727270d_enat @ X5 @ Y6 ) ) ) ) ) ).

% finite_enum_subset
thf(fact_1415_finite__enum__subset,axiom,
    ! [X5: set_nat,Y6: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( finite_card_nat @ X5 ) )
         => ( ( infini8530281810654367211te_nat @ X5 @ I2 )
            = ( infini8530281810654367211te_nat @ Y6 @ I2 ) ) )
     => ( ( finite_finite_nat @ X5 )
       => ( ( finite_finite_nat @ Y6 )
         => ( ( ord_less_eq_nat @ ( finite_card_nat @ X5 ) @ ( finite_card_nat @ Y6 ) )
           => ( ord_less_eq_set_nat @ X5 @ Y6 ) ) ) ) ) ).

% finite_enum_subset
thf(fact_1416_inverse__of__nat__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_1417_inverse__of__nat__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_1418_Diff__cancel,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ A2 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_1419_Diff__cancel,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ A2 @ A2 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_1420_Diff__cancel,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ A2 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_1421_Diff__cancel,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ A2 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_1422_empty__Diff,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_1423_empty__Diff,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_1424_empty__Diff,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_1425_empty__Diff,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_1426_Diff__empty,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% Diff_empty
thf(fact_1427_Diff__empty,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
      = A2 ) ).

% Diff_empty
thf(fact_1428_Diff__empty,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% Diff_empty
thf(fact_1429_Diff__empty,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Diff_empty
thf(fact_1430_finite__Diff2,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) )
        = ( finite_finite_int @ A2 ) ) ) ).

% finite_Diff2
thf(fact_1431_finite__Diff2,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
        = ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_Diff2
thf(fact_1432_finite__Diff2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
        = ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_1433_finite__Diff2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A2 @ B2 ) )
        = ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_1434_finite__Diff2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
        = ( finite_finite_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_1435_finite__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).

% finite_Diff
thf(fact_1436_finite__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).

% finite_Diff
thf(fact_1437_finite__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).

% finite_Diff
thf(fact_1438_finite__Diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) ) ).

% finite_Diff
thf(fact_1439_finite__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% finite_Diff
thf(fact_1440_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1441_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1442_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1443_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1444_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_1445_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1446_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1447_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1448_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_1449_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_1450_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_1451_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_1452_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_1453_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( minus_minus_set_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_1454_Diff__eq__empty__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( minus_minus_set_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_1455_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( minus_minus_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_1456_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( minus_minus_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_1457_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_1458_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1459_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1460_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1461_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_1462_card_Oempty,axiom,
    ( ( finite_card_complex @ bot_bot_set_complex )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1463_card_Oempty,axiom,
    ( ( finite_card_list_nat @ bot_bot_set_list_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1464_card_Oempty,axiom,
    ( ( finite_card_set_nat @ bot_bot_set_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1465_card_Oempty,axiom,
    ( ( finite_card_real @ bot_bot_set_real )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1466_card_Oempty,axiom,
    ( ( finite_card_o @ bot_bot_set_o )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1467_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1468_card_Oempty,axiom,
    ( ( finite_card_int @ bot_bot_set_int )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1469_card_Oinfinite,axiom,
    ! [A2: set_list_nat] :
      ( ~ ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite_card_list_nat @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1470_card_Oinfinite,axiom,
    ! [A2: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite_card_set_nat @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1471_card_Oinfinite,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_card_nat @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1472_card_Oinfinite,axiom,
    ! [A2: set_int] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( finite_card_int @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1473_card_Oinfinite,axiom,
    ! [A2: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_card_complex @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1474_card_Oinfinite,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite711546835091564841at_nat @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1475_card_Oinfinite,axiom,
    ! [A2: set_Extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite121521170596916366d_enat @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1476_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% zle_diff1_eq
thf(fact_1477_card__0__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( ( finite_card_list_nat @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
thf(fact_1478_card__0__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( ( finite_card_set_nat @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
thf(fact_1479_card__0__eq,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( finite_card_complex @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_complex ) ) ) ).

% card_0_eq
thf(fact_1480_card__0__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ( finite711546835091564841at_nat @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bo2099793752762293965at_nat ) ) ) ).

% card_0_eq
thf(fact_1481_card__0__eq,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( finite121521170596916366d_enat @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bo7653980558646680370d_enat ) ) ) ).

% card_0_eq
thf(fact_1482_card__0__eq,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( finite_card_real @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_real ) ) ) ).

% card_0_eq
thf(fact_1483_card__0__eq,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( finite_card_o @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_o ) ) ) ).

% card_0_eq
thf(fact_1484_card__0__eq,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( finite_card_nat @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_1485_card__0__eq,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( finite_card_int @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_int ) ) ) ).

% card_0_eq
thf(fact_1486_finite__enumerate__mono__iff,axiom,
    ! [S2: set_Extended_enat,M2: nat,N: nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ord_less_nat @ M2 @ ( finite121521170596916366d_enat @ S2 ) )
       => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S2 ) )
         => ( ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ M2 ) @ ( infini7641415182203889163d_enat @ S2 @ N ) )
            = ( ord_less_nat @ M2 @ N ) ) ) ) ) ).

% finite_enumerate_mono_iff
thf(fact_1487_finite__enumerate__mono__iff,axiom,
    ! [S2: set_nat,M2: nat,N: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ord_less_nat @ M2 @ ( finite_card_nat @ S2 ) )
       => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
         => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) )
            = ( ord_less_nat @ M2 @ N ) ) ) ) ) ).

% finite_enumerate_mono_iff
thf(fact_1488_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nonneg_int_cases
thf(fact_1489_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_1490_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% int_one_le_iff_zero_less
thf(fact_1491_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_1492_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_1493_conj__le__cong,axiom,
    ! [X: int,X7: int,P: $o,P4: $o] :
      ( ( X = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P4 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
            & P4 ) ) ) ) ).

% conj_le_cong
thf(fact_1494_imp__le__cong,axiom,
    ! [X: int,X7: int,P: $o,P4: $o] :
      ( ( X = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P4 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
           => P4 ) ) ) ) ).

% imp_le_cong
thf(fact_1495_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_1496_card__less__sym__Diff,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite8100373058378681591st_nat @ B2 )
       => ( ( ord_less_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) )
         => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1497_card__less__sym__Diff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
         => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1498_card__less__sym__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( ord_less_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) )
         => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1499_card__less__sym__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ord_less_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) )
         => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1500_card__less__sym__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite6177210948735845034at_nat @ B2 )
       => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A2 ) @ ( finite711546835091564841at_nat @ B2 ) )
         => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1501_card__less__sym__Diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ord_less_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) )
         => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1502_card__less__sym__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1503_card__le__sym__Diff,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite8100373058378681591st_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1504_card__le__sym__Diff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1505_card__le__sym__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1506_card__le__sym__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1507_card__le__sym__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite6177210948735845034at_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A2 ) @ ( finite711546835091564841at_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1508_card__le__sym__Diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1509_card__le__sym__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1510_Diff__infinite__finite,axiom,
    ! [T3: set_int,S2: set_int] :
      ( ( finite_finite_int @ T3 )
     => ( ~ ( finite_finite_int @ S2 )
       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1511_Diff__infinite__finite,axiom,
    ! [T3: set_complex,S2: set_complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S2 )
       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1512_Diff__infinite__finite,axiom,
    ! [T3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ T3 )
     => ( ~ ( finite6177210948735845034at_nat @ S2 )
       => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S2 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1513_Diff__infinite__finite,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ~ ( finite4001608067531595151d_enat @ S2 )
       => ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ S2 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1514_Diff__infinite__finite,axiom,
    ! [T3: set_nat,S2: set_nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1515_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_1516_double__diff,axiom,
    ! [A2: set_int,B2: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C2 )
       => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C2 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_1517_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).

% Diff_subset
thf(fact_1518_Diff__subset,axiom,
    ! [A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ A2 ) ).

% Diff_subset
thf(fact_1519_Diff__mono,axiom,
    ! [A2: set_nat,C2: set_nat,D4: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C2 )
     => ( ( ord_less_eq_set_nat @ D4 @ B2 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_1520_Diff__mono,axiom,
    ! [A2: set_int,C2: set_int,D4: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C2 )
     => ( ( ord_less_eq_set_int @ D4 @ B2 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( minus_minus_set_int @ C2 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_1521_psubset__imp__ex__mem,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ord_less_set_o @ A2 @ B2 )
     => ? [B5: $o] : ( member_o @ B5 @ ( minus_minus_set_o @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1522_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B2 )
     => ? [B5: set_nat] : ( member_set_nat @ B5 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1523_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ( ord_le1311537459589289991at_rat @ A2 @ B2 )
     => ? [B5: set_nat_rat] : ( member_set_nat_rat @ B5 @ ( minus_1626877696091177228at_rat @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1524_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ? [B5: int] : ( member_int @ B5 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1525_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1526_card__Diff__subset,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1527_card__Diff__subset,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1528_card__Diff__subset,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1529_card__Diff__subset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ B2 @ A2 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A2 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1530_card__Diff__subset,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
       => ( ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1531_card__Diff__subset,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1532_card__Diff__subset,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1533_diff__card__le__card__Diff,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1534_diff__card__le__card__Diff,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1535_diff__card__le__card__Diff,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1536_diff__card__le__card__Diff,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1537_diff__card__le__card__Diff,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite711546835091564841at_nat @ A2 ) @ ( finite711546835091564841at_nat @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1538_diff__card__le__card__Diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1539_diff__card__le__card__Diff,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1540_card__subset__eq,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
       => ( ( ( finite_card_list_nat @ A2 )
            = ( finite_card_list_nat @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1541_card__subset__eq,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
       => ( ( ( finite_card_set_nat @ A2 )
            = ( finite_card_set_nat @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1542_card__subset__eq,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ( ( finite_card_nat @ A2 )
            = ( finite_card_nat @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1543_card__subset__eq,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( ( finite_card_complex @ A2 )
            = ( finite_card_complex @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1544_card__subset__eq,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
       => ( ( ( finite711546835091564841at_nat @ A2 )
            = ( finite711546835091564841at_nat @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1545_card__subset__eq,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ( ( finite121521170596916366d_enat @ A2 )
            = ( finite121521170596916366d_enat @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1546_card__subset__eq,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( ( finite_card_int @ A2 )
            = ( finite_card_int @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1547_infinite__arbitrarily__large,axiom,
    ! [A2: set_list_nat,N: nat] :
      ( ~ ( finite8100373058378681591st_nat @ A2 )
     => ? [B8: set_list_nat] :
          ( ( finite8100373058378681591st_nat @ B8 )
          & ( ( finite_card_list_nat @ B8 )
            = N )
          & ( ord_le6045566169113846134st_nat @ B8 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1548_infinite__arbitrarily__large,axiom,
    ! [A2: set_set_nat,N: nat] :
      ( ~ ( finite1152437895449049373et_nat @ A2 )
     => ? [B8: set_set_nat] :
          ( ( finite1152437895449049373et_nat @ B8 )
          & ( ( finite_card_set_nat @ B8 )
            = N )
          & ( ord_le6893508408891458716et_nat @ B8 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1549_infinite__arbitrarily__large,axiom,
    ! [A2: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ? [B8: set_nat] :
          ( ( finite_finite_nat @ B8 )
          & ( ( finite_card_nat @ B8 )
            = N )
          & ( ord_less_eq_set_nat @ B8 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1550_infinite__arbitrarily__large,axiom,
    ! [A2: set_complex,N: nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ? [B8: set_complex] :
          ( ( finite3207457112153483333omplex @ B8 )
          & ( ( finite_card_complex @ B8 )
            = N )
          & ( ord_le211207098394363844omplex @ B8 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1551_infinite__arbitrarily__large,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ? [B8: set_Pr1261947904930325089at_nat] :
          ( ( finite6177210948735845034at_nat @ B8 )
          & ( ( finite711546835091564841at_nat @ B8 )
            = N )
          & ( ord_le3146513528884898305at_nat @ B8 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1552_infinite__arbitrarily__large,axiom,
    ! [A2: set_Extended_enat,N: nat] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ? [B8: set_Extended_enat] :
          ( ( finite4001608067531595151d_enat @ B8 )
          & ( ( finite121521170596916366d_enat @ B8 )
            = N )
          & ( ord_le7203529160286727270d_enat @ B8 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1553_infinite__arbitrarily__large,axiom,
    ! [A2: set_int,N: nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ? [B8: set_int] :
          ( ( finite_finite_int @ B8 )
          & ( ( finite_card_int @ B8 )
            = N )
          & ( ord_less_eq_set_int @ B8 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1554_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_o,R2: $o > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ? [B9: $o] :
                ( ( member_o @ B9 @ B2 )
                & ( R2 @ A5 @ B9 ) ) )
       => ( ! [A1: $o,A22: $o,B5: $o] :
              ( ( member_o @ A1 @ A2 )
             => ( ( member_o @ A22 @ A2 )
               => ( ( member_o @ B5 @ B2 )
                 => ( ( R2 @ A1 @ B5 )
                   => ( ( R2 @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1555_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_complex,R2: complex > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ? [B9: $o] :
                ( ( member_o @ B9 @ B2 )
                & ( R2 @ A5 @ B9 ) ) )
       => ( ! [A1: complex,A22: complex,B5: $o] :
              ( ( member_complex @ A1 @ A2 )
             => ( ( member_complex @ A22 @ A2 )
               => ( ( member_o @ B5 @ B2 )
                 => ( ( R2 @ A1 @ B5 )
                   => ( ( R2 @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1556_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_nat,R2: nat > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ? [B9: $o] :
                ( ( member_o @ B9 @ B2 )
                & ( R2 @ A5 @ B9 ) ) )
       => ( ! [A1: nat,A22: nat,B5: $o] :
              ( ( member_nat @ A1 @ A2 )
             => ( ( member_nat @ A22 @ A2 )
               => ( ( member_o @ B5 @ B2 )
                 => ( ( R2 @ A1 @ B5 )
                   => ( ( R2 @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1557_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_int,R2: int > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ? [B9: $o] :
                ( ( member_o @ B9 @ B2 )
                & ( R2 @ A5 @ B9 ) ) )
       => ( ! [A1: int,A22: int,B5: $o] :
              ( ( member_int @ A1 @ A2 )
             => ( ( member_int @ A22 @ A2 )
               => ( ( member_o @ B5 @ B2 )
                 => ( ( R2 @ A1 @ B5 )
                   => ( ( R2 @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1558_card__le__if__inj__on__rel,axiom,
    ! [B2: set_nat,A2: set_o,R2: $o > nat > $o] :
      ( ( finite_finite_nat @ B2 )
     => ( ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ? [B9: nat] :
                ( ( member_nat @ B9 @ B2 )
                & ( R2 @ A5 @ B9 ) ) )
       => ( ! [A1: $o,A22: $o,B5: nat] :
              ( ( member_o @ A1 @ A2 )
             => ( ( member_o @ A22 @ A2 )
               => ( ( member_nat @ B5 @ B2 )
                 => ( ( R2 @ A1 @ B5 )
                   => ( ( R2 @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1559_card__le__if__inj__on__rel,axiom,
    ! [B2: set_nat,A2: set_complex,R2: complex > nat > $o] :
      ( ( finite_finite_nat @ B2 )
     => ( ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ? [B9: nat] :
                ( ( member_nat @ B9 @ B2 )
                & ( R2 @ A5 @ B9 ) ) )
       => ( ! [A1: complex,A22: complex,B5: nat] :
              ( ( member_complex @ A1 @ A2 )
             => ( ( member_complex @ A22 @ A2 )
               => ( ( member_nat @ B5 @ B2 )
                 => ( ( R2 @ A1 @ B5 )
                   => ( ( R2 @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1560_card__le__if__inj__on__rel,axiom,
    ! [B2: set_nat,A2: set_nat,R2: nat > nat > $o] :
      ( ( finite_finite_nat @ B2 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ? [B9: nat] :
                ( ( member_nat @ B9 @ B2 )
                & ( R2 @ A5 @ B9 ) ) )
       => ( ! [A1: nat,A22: nat,B5: nat] :
              ( ( member_nat @ A1 @ A2 )
             => ( ( member_nat @ A22 @ A2 )
               => ( ( member_nat @ B5 @ B2 )
                 => ( ( R2 @ A1 @ B5 )
                   => ( ( R2 @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1561_card__le__if__inj__on__rel,axiom,
    ! [B2: set_nat,A2: set_int,R2: int > nat > $o] :
      ( ( finite_finite_nat @ B2 )
     => ( ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ? [B9: nat] :
                ( ( member_nat @ B9 @ B2 )
                & ( R2 @ A5 @ B9 ) ) )
       => ( ! [A1: int,A22: int,B5: nat] :
              ( ( member_int @ A1 @ A2 )
             => ( ( member_int @ A22 @ A2 )
               => ( ( member_nat @ B5 @ B2 )
                 => ( ( R2 @ A1 @ B5 )
                   => ( ( R2 @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1562_card__le__if__inj__on__rel,axiom,
    ! [B2: set_int,A2: set_o,R2: $o > int > $o] :
      ( ( finite_finite_int @ B2 )
     => ( ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ? [B9: int] :
                ( ( member_int @ B9 @ B2 )
                & ( R2 @ A5 @ B9 ) ) )
       => ( ! [A1: $o,A22: $o,B5: int] :
              ( ( member_o @ A1 @ A2 )
             => ( ( member_o @ A22 @ A2 )
               => ( ( member_int @ B5 @ B2 )
                 => ( ( R2 @ A1 @ B5 )
                   => ( ( R2 @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1563_card__le__if__inj__on__rel,axiom,
    ! [B2: set_int,A2: set_complex,R2: complex > int > $o] :
      ( ( finite_finite_int @ B2 )
     => ( ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ? [B9: int] :
                ( ( member_int @ B9 @ B2 )
                & ( R2 @ A5 @ B9 ) ) )
       => ( ! [A1: complex,A22: complex,B5: int] :
              ( ( member_complex @ A1 @ A2 )
             => ( ( member_complex @ A22 @ A2 )
               => ( ( member_int @ B5 @ B2 )
                 => ( ( R2 @ A1 @ B5 )
                   => ( ( R2 @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1564_set__encode__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ( nat_set_encode @ A2 )
            = ( nat_set_encode @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% set_encode_eq
thf(fact_1565_frac__ge__0,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).

% frac_ge_0
thf(fact_1566_frac__ge__0,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).

% frac_ge_0
thf(fact_1567_frac__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).

% frac_lt_1
thf(fact_1568_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_1569_card__eq__0__iff,axiom,
    ! [A2: set_list_nat] :
      ( ( ( finite_card_list_nat @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_list_nat )
        | ~ ( finite8100373058378681591st_nat @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_1570_card__eq__0__iff,axiom,
    ! [A2: set_set_nat] :
      ( ( ( finite_card_set_nat @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_set_nat )
        | ~ ( finite1152437895449049373et_nat @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_1571_card__eq__0__iff,axiom,
    ! [A2: set_complex] :
      ( ( ( finite_card_complex @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_complex )
        | ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_1572_card__eq__0__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bo2099793752762293965at_nat )
        | ~ ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_1573_card__eq__0__iff,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ( finite121521170596916366d_enat @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bo7653980558646680370d_enat )
        | ~ ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_1574_card__eq__0__iff,axiom,
    ! [A2: set_real] :
      ( ( ( finite_card_real @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_real )
        | ~ ( finite_finite_real @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_1575_card__eq__0__iff,axiom,
    ! [A2: set_o] :
      ( ( ( finite_card_o @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_o )
        | ~ ( finite_finite_o @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_1576_card__eq__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( ( finite_card_nat @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_1577_card__eq__0__iff,axiom,
    ! [A2: set_int] :
      ( ( ( finite_card_int @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_int )
        | ~ ( finite_finite_int @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_1578_card__ge__0__finite,axiom,
    ! [A2: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A2 ) )
     => ( finite8100373058378681591st_nat @ A2 ) ) ).

% card_ge_0_finite
thf(fact_1579_card__ge__0__finite,axiom,
    ! [A2: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A2 ) )
     => ( finite1152437895449049373et_nat @ A2 ) ) ).

% card_ge_0_finite
thf(fact_1580_card__ge__0__finite,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
     => ( finite_finite_nat @ A2 ) ) ).

% card_ge_0_finite
thf(fact_1581_card__ge__0__finite,axiom,
    ! [A2: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A2 ) )
     => ( finite_finite_int @ A2 ) ) ).

% card_ge_0_finite
thf(fact_1582_card__ge__0__finite,axiom,
    ! [A2: set_complex] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A2 ) )
     => ( finite3207457112153483333omplex @ A2 ) ) ).

% card_ge_0_finite
thf(fact_1583_card__ge__0__finite,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A2 ) )
     => ( finite6177210948735845034at_nat @ A2 ) ) ).

% card_ge_0_finite
thf(fact_1584_card__ge__0__finite,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite121521170596916366d_enat @ A2 ) )
     => ( finite4001608067531595151d_enat @ A2 ) ) ).

% card_ge_0_finite
thf(fact_1585_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_list_nat,C2: nat] :
      ( ! [G: set_list_nat] :
          ( ( ord_le6045566169113846134st_nat @ G @ F2 )
         => ( ( finite8100373058378681591st_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_list_nat @ G ) @ C2 ) ) )
     => ( ( finite8100373058378681591st_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_list_nat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1586_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_set_nat,C2: nat] :
      ( ! [G: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ G @ F2 )
         => ( ( finite1152437895449049373et_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_set_nat @ G ) @ C2 ) ) )
     => ( ( finite1152437895449049373et_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_set_nat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1587_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_nat,C2: nat] :
      ( ! [G: set_nat] :
          ( ( ord_less_eq_set_nat @ G @ F2 )
         => ( ( finite_finite_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
     => ( ( finite_finite_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1588_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_complex,C2: nat] :
      ( ! [G: set_complex] :
          ( ( ord_le211207098394363844omplex @ G @ F2 )
         => ( ( finite3207457112153483333omplex @ G )
           => ( ord_less_eq_nat @ ( finite_card_complex @ G ) @ C2 ) ) )
     => ( ( finite3207457112153483333omplex @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_complex @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1589_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ! [G: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ G @ F2 )
         => ( ( finite6177210948735845034at_nat @ G )
           => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ G ) @ C2 ) ) )
     => ( ( finite6177210948735845034at_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1590_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Extended_enat,C2: nat] :
      ( ! [G: set_Extended_enat] :
          ( ( ord_le7203529160286727270d_enat @ G @ F2 )
         => ( ( finite4001608067531595151d_enat @ G )
           => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ G ) @ C2 ) ) )
     => ( ( finite4001608067531595151d_enat @ F2 )
        & ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1591_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_int,C2: nat] :
      ( ! [G: set_int] :
          ( ( ord_less_eq_set_int @ G @ F2 )
         => ( ( finite_finite_int @ G )
           => ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C2 ) ) )
     => ( ( finite_finite_int @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_int @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1592_obtain__subset__with__card__n,axiom,
    ! [N: nat,S2: set_list_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ S2 ) )
     => ~ ! [T4: set_list_nat] :
            ( ( ord_le6045566169113846134st_nat @ T4 @ S2 )
           => ( ( ( finite_card_list_nat @ T4 )
                = N )
             => ~ ( finite8100373058378681591st_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1593_obtain__subset__with__card__n,axiom,
    ! [N: nat,S2: set_set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S2 ) )
     => ~ ! [T4: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ T4 @ S2 )
           => ( ( ( finite_card_set_nat @ T4 )
                = N )
             => ~ ( finite1152437895449049373et_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1594_obtain__subset__with__card__n,axiom,
    ! [N: nat,S2: set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
     => ~ ! [T4: set_nat] :
            ( ( ord_less_eq_set_nat @ T4 @ S2 )
           => ( ( ( finite_card_nat @ T4 )
                = N )
             => ~ ( finite_finite_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1595_obtain__subset__with__card__n,axiom,
    ! [N: nat,S2: set_complex] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_complex @ S2 ) )
     => ~ ! [T4: set_complex] :
            ( ( ord_le211207098394363844omplex @ T4 @ S2 )
           => ( ( ( finite_card_complex @ T4 )
                = N )
             => ~ ( finite3207457112153483333omplex @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1596_obtain__subset__with__card__n,axiom,
    ! [N: nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ S2 ) )
     => ~ ! [T4: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ T4 @ S2 )
           => ( ( ( finite711546835091564841at_nat @ T4 )
                = N )
             => ~ ( finite6177210948735845034at_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1597_obtain__subset__with__card__n,axiom,
    ! [N: nat,S2: set_Extended_enat] :
      ( ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ S2 ) )
     => ~ ! [T4: set_Extended_enat] :
            ( ( ord_le7203529160286727270d_enat @ T4 @ S2 )
           => ( ( ( finite121521170596916366d_enat @ T4 )
                = N )
             => ~ ( finite4001608067531595151d_enat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1598_obtain__subset__with__card__n,axiom,
    ! [N: nat,S2: set_int] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_int @ S2 ) )
     => ~ ! [T4: set_int] :
            ( ( ord_less_eq_set_int @ T4 @ S2 )
           => ( ( ( finite_card_int @ T4 )
                = N )
             => ~ ( finite_finite_int @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1599_card__seteq,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B2 ) @ ( finite_card_list_nat @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1600_card__seteq,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1601_card__seteq,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1602_card__seteq,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_complex @ B2 ) @ ( finite_card_complex @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1603_card__seteq,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ B2 ) @ ( finite711546835091564841at_nat @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1604_card__seteq,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ B2 ) @ ( finite121521170596916366d_enat @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1605_card__seteq,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ B2 ) @ ( finite_card_int @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1606_card__mono,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_1607_card__mono,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_1608_card__mono,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_1609_card__mono,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) ) ) ) ).

% card_mono
thf(fact_1610_card__mono,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A2 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_1611_card__mono,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).

% card_mono
thf(fact_1612_card__mono,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ).

% card_mono
thf(fact_1613_psubset__card__mono,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le1190675801316882794st_nat @ A2 @ B2 )
       => ( ord_less_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1614_psubset__card__mono,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_less_set_set_nat @ A2 @ B2 )
       => ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1615_psubset__card__mono,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_set_nat @ A2 @ B2 )
       => ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1616_psubset__card__mono,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_set_int @ A2 @ B2 )
       => ( ord_less_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1617_psubset__card__mono,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_less_set_complex @ A2 @ B2 )
       => ( ord_less_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1618_psubset__card__mono,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
       => ( ord_less_nat @ ( finite711546835091564841at_nat @ A2 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1619_psubset__card__mono,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le2529575680413868914d_enat @ A2 @ B2 )
       => ( ord_less_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1620_finite__le__enumerate,axiom,
    ! [S2: set_nat,N: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
       => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).

% finite_le_enumerate
thf(fact_1621_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_1622_finite__enumerate__in__set,axiom,
    ! [S2: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S2 ) )
       => ( member_Extended_enat @ ( infini7641415182203889163d_enat @ S2 @ N ) @ S2 ) ) ) ).

% finite_enumerate_in_set
thf(fact_1623_finite__enumerate__in__set,axiom,
    ! [S2: set_nat,N: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
       => ( member_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ S2 ) ) ) ).

% finite_enumerate_in_set
thf(fact_1624_finite__enumerate__Ex,axiom,
    ! [S2: set_Extended_enat,S: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( member_Extended_enat @ S @ S2 )
       => ? [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( finite121521170596916366d_enat @ S2 ) )
            & ( ( infini7641415182203889163d_enat @ S2 @ N2 )
              = S ) ) ) ) ).

% finite_enumerate_Ex
thf(fact_1625_finite__enumerate__Ex,axiom,
    ! [S2: set_nat,S: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( member_nat @ S @ S2 )
       => ? [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S2 ) )
            & ( ( infini8530281810654367211te_nat @ S2 @ N2 )
              = S ) ) ) ) ).

% finite_enumerate_Ex
thf(fact_1626_finite__enum__ext,axiom,
    ! [X5: set_Extended_enat,Y6: set_Extended_enat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( finite121521170596916366d_enat @ X5 ) )
         => ( ( infini7641415182203889163d_enat @ X5 @ I2 )
            = ( infini7641415182203889163d_enat @ Y6 @ I2 ) ) )
     => ( ( finite4001608067531595151d_enat @ X5 )
       => ( ( finite4001608067531595151d_enat @ Y6 )
         => ( ( ( finite121521170596916366d_enat @ X5 )
              = ( finite121521170596916366d_enat @ Y6 ) )
           => ( X5 = Y6 ) ) ) ) ) ).

% finite_enum_ext
thf(fact_1627_finite__enum__ext,axiom,
    ! [X5: set_nat,Y6: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( finite_card_nat @ X5 ) )
         => ( ( infini8530281810654367211te_nat @ X5 @ I2 )
            = ( infini8530281810654367211te_nat @ Y6 @ I2 ) ) )
     => ( ( finite_finite_nat @ X5 )
       => ( ( finite_finite_nat @ Y6 )
         => ( ( ( finite_card_nat @ X5 )
              = ( finite_card_nat @ Y6 ) )
           => ( X5 = Y6 ) ) ) ) ) ).

% finite_enum_ext
thf(fact_1628_zle__int,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% zle_int
thf(fact_1629_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( nat_set_encode @ A2 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_1630_card__gt__0__iff,axiom,
    ! [A2: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A2 ) )
      = ( ( A2 != bot_bot_set_list_nat )
        & ( finite8100373058378681591st_nat @ A2 ) ) ) ).

% card_gt_0_iff
thf(fact_1631_card__gt__0__iff,axiom,
    ! [A2: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A2 ) )
      = ( ( A2 != bot_bot_set_set_nat )
        & ( finite1152437895449049373et_nat @ A2 ) ) ) ).

% card_gt_0_iff
thf(fact_1632_card__gt__0__iff,axiom,
    ! [A2: set_complex] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A2 ) )
      = ( ( A2 != bot_bot_set_complex )
        & ( finite3207457112153483333omplex @ A2 ) ) ) ).

% card_gt_0_iff
thf(fact_1633_card__gt__0__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A2 ) )
      = ( ( A2 != bot_bo2099793752762293965at_nat )
        & ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% card_gt_0_iff
thf(fact_1634_card__gt__0__iff,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite121521170596916366d_enat @ A2 ) )
      = ( ( A2 != bot_bo7653980558646680370d_enat )
        & ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% card_gt_0_iff
thf(fact_1635_card__gt__0__iff,axiom,
    ! [A2: set_real] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
      = ( ( A2 != bot_bot_set_real )
        & ( finite_finite_real @ A2 ) ) ) ).

% card_gt_0_iff
thf(fact_1636_card__gt__0__iff,axiom,
    ! [A2: set_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A2 ) )
      = ( ( A2 != bot_bot_set_o )
        & ( finite_finite_o @ A2 ) ) ) ).

% card_gt_0_iff
thf(fact_1637_card__gt__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
      = ( ( A2 != bot_bot_set_nat )
        & ( finite_finite_nat @ A2 ) ) ) ).

% card_gt_0_iff
thf(fact_1638_card__gt__0__iff,axiom,
    ! [A2: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A2 ) )
      = ( ( A2 != bot_bot_set_int )
        & ( finite_finite_int @ A2 ) ) ) ).

% card_gt_0_iff
thf(fact_1639_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X3: list_nat] :
              ( ( member_list_nat @ X3 @ A2 )
             => ! [Y2: list_nat] :
                  ( ( member_list_nat @ Y2 @ A2 )
                 => ( X3 = Y2 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1640_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X3: set_nat] :
              ( ( member_set_nat @ X3 @ A2 )
             => ! [Y2: set_nat] :
                  ( ( member_set_nat @ Y2 @ A2 )
                 => ( X3 = Y2 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1641_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ! [Y2: nat] :
                  ( ( member_nat @ Y2 @ A2 )
                 => ( X3 = Y2 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1642_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ! [Y2: int] :
                  ( ( member_int @ Y2 @ A2 )
                 => ( X3 = Y2 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1643_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ! [Y2: complex] :
                  ( ( member_complex @ Y2 @ A2 )
                 => ( X3 = Y2 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1644_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A2 )
             => ! [Y2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y2 @ A2 )
                 => ( X3 = Y2 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1645_card__le__Suc0__iff__eq,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A2 )
             => ! [Y2: extended_enat] :
                  ( ( member_Extended_enat @ Y2 @ A2 )
                 => ( X3 = Y2 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1646_card__psubset,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
       => ( ( ord_less_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) )
         => ( ord_le1190675801316882794st_nat @ A2 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1647_card__psubset,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
       => ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
         => ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1648_card__psubset,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
         => ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1649_card__psubset,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( ord_less_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) )
         => ( ord_less_set_complex @ A2 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1650_card__psubset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
       => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A2 ) @ ( finite711546835091564841at_nat @ B2 ) )
         => ( ord_le7866589430770878221at_nat @ A2 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1651_card__psubset,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ( ord_less_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) )
         => ( ord_le2529575680413868914d_enat @ A2 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1652_card__psubset,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( ord_less_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) )
         => ( ord_less_set_int @ A2 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1653_finite__enumerate__mono,axiom,
    ! [M2: nat,N: nat,S2: set_Extended_enat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( finite4001608067531595151d_enat @ S2 )
       => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S2 ) )
         => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ M2 ) @ ( infini7641415182203889163d_enat @ S2 @ N ) ) ) ) ) ).

% finite_enumerate_mono
thf(fact_1654_finite__enumerate__mono,axiom,
    ! [M2: nat,N: nat,S2: set_nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( finite_finite_nat @ S2 )
       => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
         => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ) ).

% finite_enumerate_mono
thf(fact_1655_nat__approx__posE,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ~ ! [N2: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_1656_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E2 )
     => ~ ! [N2: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_1657_finite__enumerate__step,axiom,
    ! [S2: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ord_less_nat @ ( suc @ N ) @ ( finite121521170596916366d_enat @ S2 ) )
       => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S2 @ N ) @ ( infini7641415182203889163d_enat @ S2 @ ( suc @ N ) ) ) ) ) ).

% finite_enumerate_step
thf(fact_1658_finite__enumerate__step,axiom,
    ! [S2: set_nat,N: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S2 ) )
       => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ ( infini8530281810654367211te_nat @ S2 @ ( suc @ N ) ) ) ) ) ).

% finite_enumerate_step
thf(fact_1659_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1660_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_1661_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1662_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_1663_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1664_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_1665_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1666_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_1667_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_1668_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_1669_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_1670_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1671_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_1672_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1673_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_1674_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1675_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_1676_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1677_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_1678_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_1679_Diff__idemp,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
      = ( minus_minus_set_nat @ A2 @ B2 ) ) ).

% Diff_idemp
thf(fact_1680_Diff__iff,axiom,
    ! [C: $o,A2: set_o,B2: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B2 ) )
      = ( ( member_o @ C @ A2 )
        & ~ ( member_o @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_1681_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
      = ( ( member_set_nat @ C @ A2 )
        & ~ ( member_set_nat @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_1682_Diff__iff,axiom,
    ! [C: set_nat_rat,A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ C @ ( minus_1626877696091177228at_rat @ A2 @ B2 ) )
      = ( ( member_set_nat_rat @ C @ A2 )
        & ~ ( member_set_nat_rat @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_1683_Diff__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
      = ( ( member_int @ C @ A2 )
        & ~ ( member_int @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_1684_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
      = ( ( member_nat @ C @ A2 )
        & ~ ( member_nat @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_1685_DiffI,axiom,
    ! [C: $o,A2: set_o,B2: set_o] :
      ( ( member_o @ C @ A2 )
     => ( ~ ( member_o @ C @ B2 )
       => ( member_o @ C @ ( minus_minus_set_o @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_1686_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ A2 )
     => ( ~ ( member_set_nat @ C @ B2 )
       => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_1687_DiffI,axiom,
    ! [C: set_nat_rat,A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ C @ A2 )
     => ( ~ ( member_set_nat_rat @ C @ B2 )
       => ( member_set_nat_rat @ C @ ( minus_1626877696091177228at_rat @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_1688_DiffI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ A2 )
     => ( ~ ( member_int @ C @ B2 )
       => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_1689_DiffI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( ~ ( member_nat @ C @ B2 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_1690_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_1691_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_1692_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_1693_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_1694_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_1695_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_1696_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_1697_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_1698_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = one_one_complex )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_1699_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_1700_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = one_one_real )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_1701_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A @ B ) )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_1702_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_1703_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B ) )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_1704_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_1705_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1706_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_1707_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = one_one_complex ) ) ) ).

% divide_self_if
thf(fact_1708_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_1709_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_1710_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_1711_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ( divide_divide_real @ B @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_1712_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_1713_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_1714_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_1715_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( divide_divide_real @ one_one_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_1716_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_1717_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_1718_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% divide_le_0_1_iff
thf(fact_1719_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_1720_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_1721_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_1722_DiffD2,axiom,
    ! [C: $o,A2: set_o,B2: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B2 ) )
     => ~ ( member_o @ C @ B2 ) ) ).

% DiffD2
thf(fact_1723_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
     => ~ ( member_set_nat @ C @ B2 ) ) ).

% DiffD2
thf(fact_1724_DiffD2,axiom,
    ! [C: set_nat_rat,A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ C @ ( minus_1626877696091177228at_rat @ A2 @ B2 ) )
     => ~ ( member_set_nat_rat @ C @ B2 ) ) ).

% DiffD2
thf(fact_1725_DiffD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
     => ~ ( member_int @ C @ B2 ) ) ).

% DiffD2
thf(fact_1726_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
     => ~ ( member_nat @ C @ B2 ) ) ).

% DiffD2
thf(fact_1727_DiffD1,axiom,
    ! [C: $o,A2: set_o,B2: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B2 ) )
     => ( member_o @ C @ A2 ) ) ).

% DiffD1
thf(fact_1728_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
     => ( member_set_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_1729_DiffD1,axiom,
    ! [C: set_nat_rat,A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ C @ ( minus_1626877696091177228at_rat @ A2 @ B2 ) )
     => ( member_set_nat_rat @ C @ A2 ) ) ).

% DiffD1
thf(fact_1730_DiffD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
     => ( member_int @ C @ A2 ) ) ).

% DiffD1
thf(fact_1731_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
     => ( member_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_1732_DiffE,axiom,
    ! [C: $o,A2: set_o,B2: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B2 ) )
     => ~ ( ( member_o @ C @ A2 )
         => ( member_o @ C @ B2 ) ) ) ).

% DiffE
thf(fact_1733_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
     => ~ ( ( member_set_nat @ C @ A2 )
         => ( member_set_nat @ C @ B2 ) ) ) ).

% DiffE
thf(fact_1734_DiffE,axiom,
    ! [C: set_nat_rat,A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ C @ ( minus_1626877696091177228at_rat @ A2 @ B2 ) )
     => ~ ( ( member_set_nat_rat @ C @ A2 )
         => ( member_set_nat_rat @ C @ B2 ) ) ) ).

% DiffE
thf(fact_1735_DiffE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
     => ~ ( ( member_int @ C @ A2 )
         => ( member_int @ C @ B2 ) ) ) ).

% DiffE
thf(fact_1736_DiffE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
     => ~ ( ( member_nat @ C @ A2 )
         => ( member_nat @ C @ B2 ) ) ) ).

% DiffE
thf(fact_1737_linordered__field__no__lb,axiom,
    ! [X2: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).

% linordered_field_no_lb
thf(fact_1738_linordered__field__no__lb,axiom,
    ! [X2: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X2 ) ).

% linordered_field_no_lb
thf(fact_1739_linordered__field__no__ub,axiom,
    ! [X2: real] :
    ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1740_linordered__field__no__ub,axiom,
    ! [X2: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1741_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_1742_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_1743_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_1744_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_1745_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_1746_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_1747_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_1748_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_1749_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_1750_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_1751_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_1752_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_1753_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_1754_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_1755_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_1756_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_1757_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_1758_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_1759_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_1760_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_1761_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_1762_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_1763_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_1764_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_1765_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_1766_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_1767_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_1768_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_1769_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_1770_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_1771_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_1772_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_1773_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_1774_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_1775_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = one_one_complex )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_1776_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_1777_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B )
          = one_one_real )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_1778_frac__le,axiom,
    ! [Y: real,X: real,W2: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_eq_real @ W2 @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_1779_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_1780_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_eq_real @ W2 @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_1781_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_1782_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_real @ W2 @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_1783_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_rat @ W2 @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_1784_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_1785_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_1786_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_1787_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_1788_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_1789_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_1790_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_1791_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_1792_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_1793_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_1794_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_1795_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_1796_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_1797_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_1798_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_1799_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_1800_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_1801_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_1802_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_1803_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_1804_le__div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( divide_divide_nat @ M2 @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_1805_div__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( divide_divide_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1806_div__by__Suc__0,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = M2 ) ).

% div_by_Suc_0
thf(fact_1807_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_1808_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_1809_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_1810_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_1811_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_1812_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_1813_div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( ( divide_divide_nat @ M2 @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_1814_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N4: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M3 @ N4 )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N4 ) @ N4 ) ) ) ) ) ).

% div_if
thf(fact_1815_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).

% real_of_nat_div2
thf(fact_1816_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_1817_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% real_of_nat_div4
thf(fact_1818_div__le__dividend,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).

% div_le_dividend
thf(fact_1819_div__le__mono,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).

% div_le_mono
thf(fact_1820_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( divide_divide_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M2 @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_1821_Suc__div__le__mono,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_div_le_mono
thf(fact_1822_div__greater__zero__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ord_less_eq_nat @ N @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_1823_div__le__mono2,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).

% div_le_mono2
thf(fact_1824_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ( divide_divide_nat @ M2 @ N )
          = M2 )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_1825_div__less__dividend,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).

% div_less_dividend
thf(fact_1826_zdiv__mono1,axiom,
    ! [A: int,A7: int,B: int] :
      ( ( ord_less_eq_int @ A @ A7 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A7 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_1827_zdiv__mono2,axiom,
    ! [A: int,B7: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B7 )
       => ( ( ord_less_eq_int @ B7 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B7 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_1828_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( divide_divide_int @ I @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_1829_zdiv__mono1__neg,axiom,
    ! [A: int,A7: int,B: int] :
      ( ( ord_less_eq_int @ A @ A7 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A7 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_1830_zdiv__mono2__neg,axiom,
    ! [A: int,B7: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B7 )
       => ( ( ord_less_eq_int @ B7 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B7 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_1831_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_1832_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_1833_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_1834_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_1835_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
        = ( ord_less_eq_int @ K @ I ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_1836_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_1837_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_1838_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_1839_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_1840_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_1841_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_1842_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_1843_int__power__div__base,axiom,
    ! [M2: nat,K: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ( divide_divide_int @ ( power_power_int @ K @ M2 ) @ K )
          = ( power_power_int @ K @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_1844_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_1845_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_1846_frac__unique__iff,axiom,
    ! [X: real,A: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = A )
      = ( ( member_real @ ( minus_minus_real @ X @ A ) @ ring_1_Ints_real )
        & ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ord_less_real @ A @ one_one_real ) ) ) ).

% frac_unique_iff
thf(fact_1847_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_1848_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_1849_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_real @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_real @ ( insert_real @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_1850_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_o,X: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_o @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_o @ ( insert_o @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_1851_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_complex,X: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_complex @ ( insert_complex @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_1852_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_list_nat,X: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_1853_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_set_nat,X: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( insert_set_nat @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_1854_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_1855_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_int,X: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_1856_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ( ord_less_eq_nat @ M2 @ I2 )
            & ( ord_less_nat @ I2 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( ord_less_eq_int @ ( F @ M2 ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
           => ? [I2: nat] :
                ( ( ord_less_eq_nat @ M2 @ I2 )
                & ( ord_less_eq_nat @ I2 @ N )
                & ( ( F @ I2 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_1857_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% one_less_nat_eq
thf(fact_1858_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A2 ) )
        = A2 ) ) ).

% set_encode_inverse
thf(fact_1859_enumerate__Suc_H,axiom,
    ! [S2: set_nat,N: nat] :
      ( ( infini8530281810654367211te_nat @ S2 @ ( suc @ N ) )
      = ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ ( infini8530281810654367211te_nat @ S2 @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N ) ) ).

% enumerate_Suc'
thf(fact_1860_split__div_H,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q3: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M2 )
            & ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
            & ( P @ Q3 ) ) ) ) ).

% split_div'
thf(fact_1861_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_1862_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_1863_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_1864_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_1865_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_1866_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_1867_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_1868_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_1869_insert__absorb2,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ X @ A2 ) )
      = ( insert8211810215607154385at_nat @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_1870_insert__absorb2,axiom,
    ! [X: real,A2: set_real] :
      ( ( insert_real @ X @ ( insert_real @ X @ A2 ) )
      = ( insert_real @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_1871_insert__absorb2,axiom,
    ! [X: $o,A2: set_o] :
      ( ( insert_o @ X @ ( insert_o @ X @ A2 ) )
      = ( insert_o @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_1872_insert__absorb2,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
      = ( insert_nat @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_1873_insert__absorb2,axiom,
    ! [X: int,A2: set_int] :
      ( ( insert_int @ X @ ( insert_int @ X @ A2 ) )
      = ( insert_int @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_1874_insert__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
      = ( ( A = B )
        | ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_1875_insert__iff,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
      = ( ( A = B )
        | ( member_real @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_1876_insert__iff,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A2 ) )
      = ( ( A = B )
        | ( member_o @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_1877_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
      = ( ( A = B )
        | ( member_set_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_1878_insert__iff,axiom,
    ! [A: set_nat_rat,B: set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ A @ ( insert_set_nat_rat @ B @ A2 ) )
      = ( ( A = B )
        | ( member_set_nat_rat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_1879_insert__iff,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
      = ( ( A = B )
        | ( member_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_1880_insert__iff,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
      = ( ( A = B )
        | ( member_int @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_1881_insertCI,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ A @ B2 )
       => ( A = B ) )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).

% insertCI
thf(fact_1882_insertCI,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( ~ ( member_real @ A @ B2 )
       => ( A = B ) )
     => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).

% insertCI
thf(fact_1883_insertCI,axiom,
    ! [A: $o,B2: set_o,B: $o] :
      ( ( ~ ( member_o @ A @ B2 )
       => ( A = B ) )
     => ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).

% insertCI
thf(fact_1884_insertCI,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( ( ~ ( member_set_nat @ A @ B2 )
       => ( A = B ) )
     => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).

% insertCI
thf(fact_1885_insertCI,axiom,
    ! [A: set_nat_rat,B2: set_set_nat_rat,B: set_nat_rat] :
      ( ( ~ ( member_set_nat_rat @ A @ B2 )
       => ( A = B ) )
     => ( member_set_nat_rat @ A @ ( insert_set_nat_rat @ B @ B2 ) ) ) ).

% insertCI
thf(fact_1886_insertCI,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A @ B2 )
       => ( A = B ) )
     => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).

% insertCI
thf(fact_1887_insertCI,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( ~ ( member_int @ A @ B2 )
       => ( A = B ) )
     => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).

% insertCI
thf(fact_1888_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_1889_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_1890_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_1891_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_1892_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_1893_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_1894_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1895_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1896_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1897_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1898_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1899_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1900_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1901_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1902_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_1903_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_1904_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_1905_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_1906_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1907_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_1908_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_1909_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_1910_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1911_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_1912_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_1913_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_1914_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1915_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_1916_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_1917_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_1918_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1919_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_1920_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_1921_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_1922_add_Oright__neutral,axiom,
    ! [A: literal] :
      ( ( plus_plus_literal @ A @ zero_zero_literal )
      = A ) ).

% add.right_neutral
thf(fact_1923_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_1924_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_1925_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_1926_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_1927_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_1928_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_1929_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_1930_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( ( plus_plus_real @ B @ A )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_1931_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_1932_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_1933_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_1934_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_1935_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_1936_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_1937_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_1938_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ B @ A ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_1939_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_1940_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_1941_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_1942_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ A @ B ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_1943_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_1944_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_1945_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_1946_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_1947_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_1948_add__0,axiom,
    ! [A: literal] :
      ( ( plus_plus_literal @ zero_zero_literal @ A )
      = A ) ).

% add_0
thf(fact_1949_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_1950_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_1951_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_1952_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_1953_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( plus_plus_real @ A @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% double_eq_0_iff
thf(fact_1954_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_1955_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_1956_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_1957_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_1958_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_1959_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_1960_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_1961_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_1962_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_1963_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_1964_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_1965_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_1966_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_1967_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_1968_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_1969_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_1970_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_1971_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_1972_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_1973_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_1974_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1975_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_1976_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_1977_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_1978_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1979_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_1980_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_1981_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_1982_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1983_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_1984_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_1985_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_1986_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1987_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_1988_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_1989_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_1990_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1991_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1992_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1993_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1994_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1995_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1996_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_1997_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_1998_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_1999_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_2000_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_2001_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_2002_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_2003_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_2004_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_2005_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_2006_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_2007_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_2008_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_2009_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_2010_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_2011_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_add
thf(fact_2012_abs__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
      = ( times_times_real @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_2013_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_2014_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_2015_abs__add__abs,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_add_abs
thf(fact_2016_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_2017_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_2018_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_2019_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_2020_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_2021_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_2022_singletonI,axiom,
    ! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singletonI
thf(fact_2023_singletonI,axiom,
    ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_2024_singletonI,axiom,
    ! [A: set_nat_rat] : ( member_set_nat_rat @ A @ ( insert_set_nat_rat @ A @ bot_bo6797373522285170759at_rat ) ) ).

% singletonI
thf(fact_2025_singletonI,axiom,
    ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_2026_singletonI,axiom,
    ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_2027_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_2028_singletonI,axiom,
    ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_2029_finite__insert,axiom,
    ! [A: real,A2: set_real] :
      ( ( finite_finite_real @ ( insert_real @ A @ A2 ) )
      = ( finite_finite_real @ A2 ) ) ).

% finite_insert
thf(fact_2030_finite__insert,axiom,
    ! [A: $o,A2: set_o] :
      ( ( finite_finite_o @ ( insert_o @ A @ A2 ) )
      = ( finite_finite_o @ A2 ) ) ).

% finite_insert
thf(fact_2031_finite__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
      = ( finite_finite_nat @ A2 ) ) ).

% finite_insert
thf(fact_2032_finite__insert,axiom,
    ! [A: int,A2: set_int] :
      ( ( finite_finite_int @ ( insert_int @ A @ A2 ) )
      = ( finite_finite_int @ A2 ) ) ).

% finite_insert
thf(fact_2033_finite__insert,axiom,
    ! [A: complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) )
      = ( finite3207457112153483333omplex @ A2 ) ) ).

% finite_insert
thf(fact_2034_finite__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) )
      = ( finite6177210948735845034at_nat @ A2 ) ) ).

% finite_insert
thf(fact_2035_finite__insert,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ ( insert_Extended_enat @ A @ A2 ) )
      = ( finite4001608067531595151d_enat @ A2 ) ) ).

% finite_insert
thf(fact_2036_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_2037_insert__subset,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
      = ( ( member8440522571783428010at_nat @ X @ B2 )
        & ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_2038_insert__subset,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B2 )
      = ( ( member_real @ X @ B2 )
        & ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_2039_insert__subset,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X @ A2 ) @ B2 )
      = ( ( member_o @ X @ B2 )
        & ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_2040_insert__subset,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
      = ( ( member_set_nat @ X @ B2 )
        & ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_2041_insert__subset,axiom,
    ! [X: set_nat_rat,A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ( ord_le4375437777232675859at_rat @ ( insert_set_nat_rat @ X @ A2 ) @ B2 )
      = ( ( member_set_nat_rat @ X @ B2 )
        & ( ord_le4375437777232675859at_rat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_2042_insert__subset,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
      = ( ( member_nat @ X @ B2 )
        & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_2043_insert__subset,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B2 )
      = ( ( member_int @ X @ B2 )
        & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_2044_mult__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_2045_mult__0__right,axiom,
    ! [M2: nat] :
      ( ( times_times_nat @ M2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_2046_mult__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M2 )
        = ( times_times_nat @ K @ N ) )
      = ( ( M2 = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_2047_mult__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M2 = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_2048_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_2049_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_2050_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_2051_insert__Diff1,axiom,
    ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ B2 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
        = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2052_insert__Diff1,axiom,
    ! [X: real,B2: set_real,A2: set_real] :
      ( ( member_real @ X @ B2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
        = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2053_insert__Diff1,axiom,
    ! [X: $o,B2: set_o,A2: set_o] :
      ( ( member_o @ X @ B2 )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
        = ( minus_minus_set_o @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2054_insert__Diff1,axiom,
    ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ X @ B2 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
        = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2055_insert__Diff1,axiom,
    ! [X: set_nat_rat,B2: set_set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ X @ B2 )
     => ( ( minus_1626877696091177228at_rat @ ( insert_set_nat_rat @ X @ A2 ) @ B2 )
        = ( minus_1626877696091177228at_rat @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2056_insert__Diff1,axiom,
    ! [X: int,B2: set_int,A2: set_int] :
      ( ( member_int @ X @ B2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
        = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2057_insert__Diff1,axiom,
    ! [X: nat,B2: set_nat,A2: set_nat] :
      ( ( member_nat @ X @ B2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
        = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2058_Diff__insert0,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
        = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2059_Diff__insert0,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B2 ) )
        = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2060_Diff__insert0,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ~ ( member_o @ X @ A2 )
     => ( ( minus_minus_set_o @ A2 @ ( insert_o @ X @ B2 ) )
        = ( minus_minus_set_o @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2061_Diff__insert0,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
        = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2062_Diff__insert0,axiom,
    ! [X: set_nat_rat,A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ~ ( member_set_nat_rat @ X @ A2 )
     => ( ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ B2 ) )
        = ( minus_1626877696091177228at_rat @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2063_Diff__insert0,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B2 ) )
        = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2064_Diff__insert0,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
        = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2065_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = one_one_nat )
      = ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_2066_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M2 @ N ) )
      = ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_2067_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2068_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_2069_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_2070_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2071_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_2072_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_2073_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel2
thf(fact_2074_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_2075_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_2076_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_2077_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel1
thf(fact_2078_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_2079_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_2080_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_2081_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_2082_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_2083_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_2084_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_2085_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_2086_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_2087_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_2088_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_2089_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_2090_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_2091_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_2092_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_2093_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_2094_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_2095_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_2096_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_2097_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel1
thf(fact_2098_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_2099_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_2100_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_2101_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel2
thf(fact_2102_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_2103_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_2104_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_2105_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2106_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_2107_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_2108_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2109_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_2110_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_2111_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_2112_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_2113_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_2114_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_2115_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( ( times_times_complex @ C @ A )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
thf(fact_2116_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ( times_times_real @ C @ A )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_2117_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_2118_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_2119_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_2120_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_2121_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_2122_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_2123_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( ( times_times_complex @ A @ C )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
thf(fact_2124_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ( times_times_real @ A @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_2125_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_2126_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_2127_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_2128_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = ( divide_divide_real @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2129_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_2130_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2131_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_2132_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2133_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_2134_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2135_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_2136_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2137_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_2138_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_2139_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_2140_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2141_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_2142_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_2143_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_2144_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2145_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_2146_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_2147_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_2148_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_2149_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_2150_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_2151_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_2152_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_2153_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_2154_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_2155_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_2156_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_2157_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_2158_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_2159_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_2160_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_2161_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_2162_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_2163_abs__le__self__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% abs_le_self_iff
thf(fact_2164_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_2165_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_2166_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_2167_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_2168_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_2169_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
thf(fact_2170_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_2171_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_2172_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ A2 )
        = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
      = ( ( A = B )
        & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2173_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( ( insert_real @ A @ A2 )
        = ( insert_real @ B @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2174_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A2: set_o,B: $o] :
      ( ( ( insert_o @ A @ A2 )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2175_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A2 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2176_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( ( insert_int @ A @ A2 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2177_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2178_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( ( insert_real @ B @ bot_bot_set_real )
        = ( insert_real @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2179_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A2: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2180_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2181_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2182_one__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M2 @ N ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_2183_mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_2184_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_2185_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_2186_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_2187_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_2188_mult__less__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M2 @ N ) ) ) ).

% mult_less_cancel2
thf(fact_2189_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_2190_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_iff
thf(fact_2191_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_2192_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( insert8211810215607154385at_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_2193_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
      = ( insert_real @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_2194_insert__Diff__single,axiom,
    ! [A: $o,A2: set_o] :
      ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
      = ( insert_o @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_2195_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
      = ( insert_int @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_2196_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_2197_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) ) )
      = ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2198_finite__Diff__insert,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      ( ( finite_finite_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) ) )
      = ( finite_finite_o @ ( minus_minus_set_o @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2199_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) ) )
      = ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2200_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B2 ) ) )
      = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2201_finite__Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) )
      = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2202_finite__Diff__insert,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ B2 ) ) )
      = ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2203_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2204_frac__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = zero_zero_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% frac_eq_0_iff
thf(fact_2205_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_2206_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_2207_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_2208_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_2209_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_2210_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_2211_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_2212_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_2213_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_2214_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_2215_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2216_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_2217_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2218_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2219_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_2220_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2221_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A @ zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_2222_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_2223_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_2224_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_2225_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M2 ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_2226_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_2227_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_2228_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M2 ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_2229_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M2 ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_2230_one__le__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_2231_card__insert__disjoint,axiom,
    ! [A2: set_real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( finite_card_real @ ( insert_real @ X @ A2 ) )
          = ( suc @ ( finite_card_real @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2232_card__insert__disjoint,axiom,
    ! [A2: set_o,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( finite_card_o @ ( insert_o @ X @ A2 ) )
          = ( suc @ ( finite_card_o @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2233_card__insert__disjoint,axiom,
    ! [A2: set_set_nat_rat,X: set_nat_rat] :
      ( ( finite6430367030675640852at_rat @ A2 )
     => ( ~ ( member_set_nat_rat @ X @ A2 )
       => ( ( finite8736671560171388117at_rat @ ( insert_set_nat_rat @ X @ A2 ) )
          = ( suc @ ( finite8736671560171388117at_rat @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2234_card__insert__disjoint,axiom,
    ! [A2: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ~ ( member_list_nat @ X @ A2 )
       => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A2 ) )
          = ( suc @ ( finite_card_list_nat @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2235_card__insert__disjoint,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ~ ( member_set_nat @ X @ A2 )
       => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A2 ) )
          = ( suc @ ( finite_card_set_nat @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2236_card__insert__disjoint,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
          = ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2237_card__insert__disjoint,axiom,
    ! [A2: set_int,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( finite_card_int @ ( insert_int @ X @ A2 ) )
          = ( suc @ ( finite_card_int @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2238_card__insert__disjoint,axiom,
    ! [A2: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( finite_card_complex @ ( insert_complex @ X @ A2 ) )
          = ( suc @ ( finite_card_complex @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2239_card__insert__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ~ ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) )
          = ( suc @ ( finite711546835091564841at_nat @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2240_card__insert__disjoint,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ~ ( member_Extended_enat @ X @ A2 )
       => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X @ A2 ) )
          = ( suc @ ( finite121521170596916366d_enat @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2241_mult__le__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% mult_le_cancel2
thf(fact_2242_div__mult__self__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
        = M2 ) ) ).

% div_mult_self_is_m
thf(fact_2243_div__mult__self1__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
        = M2 ) ) ).

% div_mult_self1_is_m
thf(fact_2244_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_2245_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_2246_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_2247_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W2 @ Z ) ) ) ).

% zless_nat_conj
thf(fact_2248_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% zle_add1_eq_le
thf(fact_2249_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = Z ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_2250_frac__gt__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
      = ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).

% frac_gt_0_iff
thf(fact_2251_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_2252_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% zero_less_nat_eq
thf(fact_2253_card__Diff__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ( ~ ( member8440522571783428010at_nat @ A @ B2 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2254_card__Diff__insert,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ~ ( member_real @ A @ B2 )
       => ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2255_card__Diff__insert,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ~ ( member_o @ A @ B2 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_o @ ( minus_minus_set_o @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2256_card__Diff__insert,axiom,
    ! [A: set_nat_rat,A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ A @ A2 )
     => ( ~ ( member_set_nat_rat @ A @ B2 )
       => ( ( finite8736671560171388117at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ A @ B2 ) ) )
          = ( minus_minus_nat @ ( finite8736671560171388117at_rat @ ( minus_1626877696091177228at_rat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2257_card__Diff__insert,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ A @ A2 )
     => ( ~ ( member_complex @ A @ B2 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2258_card__Diff__insert,axiom,
    ! [A: list_nat,A2: set_list_nat,B2: set_list_nat] :
      ( ( member_list_nat @ A @ A2 )
     => ( ~ ( member_list_nat @ A @ B2 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2259_card__Diff__insert,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ A @ A2 )
     => ( ~ ( member_set_nat @ A @ B2 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2260_card__Diff__insert,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ~ ( member_int @ A @ B2 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2261_card__Diff__insert,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ~ ( member_nat @ A @ B2 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2262_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).

% ln_diff_le
thf(fact_2263_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_2264_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_2265_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_2266_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_2267_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_2268_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_2269_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2270_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2271_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2272_square__diff__square__factored,axiom,
    ! [X: real,Y: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
      = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_2273_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_2274_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_2275_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_real @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% less_eq_real_def
thf(fact_2276_ln__bound,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_bound
thf(fact_2277_ln__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_ge_zero
thf(fact_2278_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_2279_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_2280_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_2281_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_2282_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X2: real] : ( member_real @ X2 @ S2 )
     => ( ? [Z5: real] :
          ! [X4: real] :
            ( ( member_real @ X4 @ S2 )
           => ( ord_less_eq_real @ X4 @ Z5 ) )
       => ? [Y3: real] :
            ( ! [X2: real] :
                ( ( member_real @ X2 @ S2 )
               => ( ord_less_eq_real @ X2 @ Y3 ) )
            & ! [Z5: real] :
                ( ! [X4: real] :
                    ( ( member_real @ X4 @ S2 )
                   => ( ord_less_eq_real @ X4 @ Z5 ) )
               => ( ord_less_eq_real @ Y3 @ Z5 ) ) ) ) ) ).

% complete_real
thf(fact_2283_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_2284_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_2285_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_2286_mk__disjoint__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ? [B8: set_Pr1261947904930325089at_nat] :
          ( ( A2
            = ( insert8211810215607154385at_nat @ A @ B8 ) )
          & ~ ( member8440522571783428010at_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2287_mk__disjoint__insert,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ? [B8: set_real] :
          ( ( A2
            = ( insert_real @ A @ B8 ) )
          & ~ ( member_real @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2288_mk__disjoint__insert,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o @ A @ A2 )
     => ? [B8: set_o] :
          ( ( A2
            = ( insert_o @ A @ B8 ) )
          & ~ ( member_o @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2289_mk__disjoint__insert,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ A2 )
     => ? [B8: set_set_nat] :
          ( ( A2
            = ( insert_set_nat @ A @ B8 ) )
          & ~ ( member_set_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2290_mk__disjoint__insert,axiom,
    ! [A: set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ A @ A2 )
     => ? [B8: set_set_nat_rat] :
          ( ( A2
            = ( insert_set_nat_rat @ A @ B8 ) )
          & ~ ( member_set_nat_rat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2291_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ? [B8: set_nat] :
          ( ( A2
            = ( insert_nat @ A @ B8 ) )
          & ~ ( member_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2292_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ? [B8: set_int] :
          ( ( A2
            = ( insert_int @ A @ B8 ) )
          & ~ ( member_int @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2293_insert__commute,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ A2 ) )
      = ( insert8211810215607154385at_nat @ Y @ ( insert8211810215607154385at_nat @ X @ A2 ) ) ) ).

% insert_commute
thf(fact_2294_insert__commute,axiom,
    ! [X: real,Y: real,A2: set_real] :
      ( ( insert_real @ X @ ( insert_real @ Y @ A2 ) )
      = ( insert_real @ Y @ ( insert_real @ X @ A2 ) ) ) ).

% insert_commute
thf(fact_2295_insert__commute,axiom,
    ! [X: $o,Y: $o,A2: set_o] :
      ( ( insert_o @ X @ ( insert_o @ Y @ A2 ) )
      = ( insert_o @ Y @ ( insert_o @ X @ A2 ) ) ) ).

% insert_commute
thf(fact_2296_insert__commute,axiom,
    ! [X: nat,Y: nat,A2: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
      = ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).

% insert_commute
thf(fact_2297_insert__commute,axiom,
    ! [X: int,Y: int,A2: set_int] :
      ( ( insert_int @ X @ ( insert_int @ Y @ A2 ) )
      = ( insert_int @ Y @ ( insert_int @ X @ A2 ) ) ) ).

% insert_commute
thf(fact_2298_insert__eq__iff,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ A2 )
     => ( ~ ( member8440522571783428010at_nat @ B @ B2 )
       => ( ( ( insert8211810215607154385at_nat @ A @ A2 )
            = ( insert8211810215607154385at_nat @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C4: set_Pr1261947904930325089at_nat] :
                  ( ( A2
                    = ( insert8211810215607154385at_nat @ B @ C4 ) )
                  & ~ ( member8440522571783428010at_nat @ B @ C4 )
                  & ( B2
                    = ( insert8211810215607154385at_nat @ A @ C4 ) )
                  & ~ ( member8440522571783428010at_nat @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2299_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B: real,B2: set_real] :
      ( ~ ( member_real @ A @ A2 )
     => ( ~ ( member_real @ B @ B2 )
       => ( ( ( insert_real @ A @ A2 )
            = ( insert_real @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C4: set_real] :
                  ( ( A2
                    = ( insert_real @ B @ C4 ) )
                  & ~ ( member_real @ B @ C4 )
                  & ( B2
                    = ( insert_real @ A @ C4 ) )
                  & ~ ( member_real @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2300_insert__eq__iff,axiom,
    ! [A: $o,A2: set_o,B: $o,B2: set_o] :
      ( ~ ( member_o @ A @ A2 )
     => ( ~ ( member_o @ B @ B2 )
       => ( ( ( insert_o @ A @ A2 )
            = ( insert_o @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A = ~ B )
             => ? [C4: set_o] :
                  ( ( A2
                    = ( insert_o @ B @ C4 ) )
                  & ~ ( member_o @ B @ C4 )
                  & ( B2
                    = ( insert_o @ A @ C4 ) )
                  & ~ ( member_o @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2301_insert__eq__iff,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ A @ A2 )
     => ( ~ ( member_set_nat @ B @ B2 )
       => ( ( ( insert_set_nat @ A @ A2 )
            = ( insert_set_nat @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C4: set_set_nat] :
                  ( ( A2
                    = ( insert_set_nat @ B @ C4 ) )
                  & ~ ( member_set_nat @ B @ C4 )
                  & ( B2
                    = ( insert_set_nat @ A @ C4 ) )
                  & ~ ( member_set_nat @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2302_insert__eq__iff,axiom,
    ! [A: set_nat_rat,A2: set_set_nat_rat,B: set_nat_rat,B2: set_set_nat_rat] :
      ( ~ ( member_set_nat_rat @ A @ A2 )
     => ( ~ ( member_set_nat_rat @ B @ B2 )
       => ( ( ( insert_set_nat_rat @ A @ A2 )
            = ( insert_set_nat_rat @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C4: set_set_nat_rat] :
                  ( ( A2
                    = ( insert_set_nat_rat @ B @ C4 ) )
                  & ~ ( member_set_nat_rat @ B @ C4 )
                  & ( B2
                    = ( insert_set_nat_rat @ A @ C4 ) )
                  & ~ ( member_set_nat_rat @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2303_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
      ( ~ ( member_nat @ A @ A2 )
     => ( ~ ( member_nat @ B @ B2 )
       => ( ( ( insert_nat @ A @ A2 )
            = ( insert_nat @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C4: set_nat] :
                  ( ( A2
                    = ( insert_nat @ B @ C4 ) )
                  & ~ ( member_nat @ B @ C4 )
                  & ( B2
                    = ( insert_nat @ A @ C4 ) )
                  & ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2304_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B: int,B2: set_int] :
      ( ~ ( member_int @ A @ A2 )
     => ( ~ ( member_int @ B @ B2 )
       => ( ( ( insert_int @ A @ A2 )
            = ( insert_int @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C4: set_int] :
                  ( ( A2
                    = ( insert_int @ B @ C4 ) )
                  & ~ ( member_int @ B @ C4 )
                  & ( B2
                    = ( insert_int @ A @ C4 ) )
                  & ~ ( member_int @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2305_insert__absorb,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ( ( insert8211810215607154385at_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_2306_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ( insert_real @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_2307_insert__absorb,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( insert_o @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_2308_insert__absorb,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ A2 )
     => ( ( insert_set_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_2309_insert__absorb,axiom,
    ! [A: set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ A @ A2 )
     => ( ( insert_set_nat_rat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_2310_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_2311_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_2312_insert__ident,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A2 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B2 )
       => ( ( ( insert8211810215607154385at_nat @ X @ A2 )
            = ( insert8211810215607154385at_nat @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2313_insert__ident,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ~ ( member_real @ X @ B2 )
       => ( ( ( insert_real @ X @ A2 )
            = ( insert_real @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2314_insert__ident,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ~ ( member_o @ X @ A2 )
     => ( ~ ( member_o @ X @ B2 )
       => ( ( ( insert_o @ X @ A2 )
            = ( insert_o @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2315_insert__ident,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ~ ( member_set_nat @ X @ B2 )
       => ( ( ( insert_set_nat @ X @ A2 )
            = ( insert_set_nat @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2316_insert__ident,axiom,
    ! [X: set_nat_rat,A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ~ ( member_set_nat_rat @ X @ A2 )
     => ( ~ ( member_set_nat_rat @ X @ B2 )
       => ( ( ( insert_set_nat_rat @ X @ A2 )
            = ( insert_set_nat_rat @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2317_insert__ident,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ~ ( member_nat @ X @ B2 )
       => ( ( ( insert_nat @ X @ A2 )
            = ( insert_nat @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2318_insert__ident,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ~ ( member_int @ X @ B2 )
       => ( ( ( insert_int @ X @ A2 )
            = ( insert_int @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2319_Set_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A2 )
     => ~ ! [B8: set_Pr1261947904930325089at_nat] :
            ( ( A2
              = ( insert8211810215607154385at_nat @ X @ B8 ) )
           => ( member8440522571783428010at_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2320_Set_Oset__insert,axiom,
    ! [X: real,A2: set_real] :
      ( ( member_real @ X @ A2 )
     => ~ ! [B8: set_real] :
            ( ( A2
              = ( insert_real @ X @ B8 ) )
           => ( member_real @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2321_Set_Oset__insert,axiom,
    ! [X: $o,A2: set_o] :
      ( ( member_o @ X @ A2 )
     => ~ ! [B8: set_o] :
            ( ( A2
              = ( insert_o @ X @ B8 ) )
           => ( member_o @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2322_Set_Oset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ X @ A2 )
     => ~ ! [B8: set_set_nat] :
            ( ( A2
              = ( insert_set_nat @ X @ B8 ) )
           => ( member_set_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2323_Set_Oset__insert,axiom,
    ! [X: set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ X @ A2 )
     => ~ ! [B8: set_set_nat_rat] :
            ( ( A2
              = ( insert_set_nat_rat @ X @ B8 ) )
           => ( member_set_nat_rat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2324_Set_Oset__insert,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( member_nat @ X @ A2 )
     => ~ ! [B8: set_nat] :
            ( ( A2
              = ( insert_nat @ X @ B8 ) )
           => ( member_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2325_Set_Oset__insert,axiom,
    ! [X: int,A2: set_int] :
      ( ( member_int @ X @ A2 )
     => ~ ! [B8: set_int] :
            ( ( A2
              = ( insert_int @ X @ B8 ) )
           => ( member_int @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2326_insertI2,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ A @ B2 )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).

% insertI2
thf(fact_2327_insertI2,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( member_real @ A @ B2 )
     => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).

% insertI2
thf(fact_2328_insertI2,axiom,
    ! [A: $o,B2: set_o,B: $o] :
      ( ( member_o @ A @ B2 )
     => ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).

% insertI2
thf(fact_2329_insertI2,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( ( member_set_nat @ A @ B2 )
     => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).

% insertI2
thf(fact_2330_insertI2,axiom,
    ! [A: set_nat_rat,B2: set_set_nat_rat,B: set_nat_rat] :
      ( ( member_set_nat_rat @ A @ B2 )
     => ( member_set_nat_rat @ A @ ( insert_set_nat_rat @ B @ B2 ) ) ) ).

% insertI2
thf(fact_2331_insertI2,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( member_nat @ A @ B2 )
     => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).

% insertI2
thf(fact_2332_insertI2,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( member_int @ A @ B2 )
     => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).

% insertI2
thf(fact_2333_insertI1,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).

% insertI1
thf(fact_2334_insertI1,axiom,
    ! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).

% insertI1
thf(fact_2335_insertI1,axiom,
    ! [A: $o,B2: set_o] : ( member_o @ A @ ( insert_o @ A @ B2 ) ) ).

% insertI1
thf(fact_2336_insertI1,axiom,
    ! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).

% insertI1
thf(fact_2337_insertI1,axiom,
    ! [A: set_nat_rat,B2: set_set_nat_rat] : ( member_set_nat_rat @ A @ ( insert_set_nat_rat @ A @ B2 ) ) ).

% insertI1
thf(fact_2338_insertI1,axiom,
    ! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).

% insertI1
thf(fact_2339_insertI1,axiom,
    ! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).

% insertI1
thf(fact_2340_insertE,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
     => ( ( A != B )
       => ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_2341_insertE,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
     => ( ( A != B )
       => ( member_real @ A @ A2 ) ) ) ).

% insertE
thf(fact_2342_insertE,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A2 ) )
     => ( ( A = ~ B )
       => ( member_o @ A @ A2 ) ) ) ).

% insertE
thf(fact_2343_insertE,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
     => ( ( A != B )
       => ( member_set_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_2344_insertE,axiom,
    ! [A: set_nat_rat,B: set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ A @ ( insert_set_nat_rat @ B @ A2 ) )
     => ( ( A != B )
       => ( member_set_nat_rat @ A @ A2 ) ) ) ).

% insertE
thf(fact_2345_insertE,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
     => ( ( A != B )
       => ( member_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_2346_insertE,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
     => ( ( A != B )
       => ( member_int @ A @ A2 ) ) ) ).

% insertE
thf(fact_2347_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2348_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2349_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2350_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2351_distrib__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% distrib_right
thf(fact_2352_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_2353_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_2354_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_2355_distrib__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_2356_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_2357_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_2358_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_2359_abs__mult,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_mult
thf(fact_2360_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_2361_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_2362_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2363_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_2364_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_2365_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_2366_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_2367_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_2368_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_2369_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2370_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_2371_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_2372_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2373_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_2374_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_2375_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_2376_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_2377_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_2378_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_2379_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_2380_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_2381_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_2382_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_2383_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2384_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_2385_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_2386_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_2387_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.left_commute
thf(fact_2388_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_2389_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_2390_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_2391_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_2392_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A4: rat,B4: rat] : ( times_times_rat @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_2393_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_2394_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_2395_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_2396_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_2397_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_2398_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_2399_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_2400_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_2401_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_2402_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2403_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_2404_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_2405_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_2406_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_2407_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_2408_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_2409_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_2410_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_2411_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_2412_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_2413_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2414_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2415_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2416_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2417_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2418_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2419_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2420_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2421_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2422_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2423_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2424_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2425_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2426_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_2427_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_2428_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_2429_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_2430_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_2431_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_2432_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_2433_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( ( plus_plus_real @ A @ A )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_2434_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_2435_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_2436_ln__le__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_2437_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
        = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_2438_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_2439_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_2440_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          | ( ord_less_eq_real @ A @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B )
          | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_2441_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_2442_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_2443_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_2444_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_2445_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_2446_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_2447_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_2448_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_2449_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
        & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_2450_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_2451_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_2452_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_2453_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_2454_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_2455_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2456_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2457_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2458_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_2459_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_2460_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_2461_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2462_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2463_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2464_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2465_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2466_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2467_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2468_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2469_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2470_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2471_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2472_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2473_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2474_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2475_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2476_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2477_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2478_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2479_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2480_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2481_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_2482_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_2483_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_2484_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2485_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2486_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2487_square__diff__one__factored,axiom,
    ! [X: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_2488_square__diff__one__factored,axiom,
    ! [X: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_2489_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_2490_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_2491_Ints__odd__nonzero,axiom,
    ! [A: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
       != zero_zero_complex ) ) ).

% Ints_odd_nonzero
thf(fact_2492_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
       != zero_zero_real ) ) ).

% Ints_odd_nonzero
thf(fact_2493_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_2494_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_2495_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_2496_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_2497_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_2498_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% abs_le_D1
thf(fact_2499_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_2500_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_2501_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_2502_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_2503_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_2504_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_2505_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_2506_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_2507_abs__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
      = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_2508_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_2509_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_2510_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2511_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_2512_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_2513_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_2514_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2515_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_2516_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_2517_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_2518_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
        ? [C5: nat] :
          ( B4
          = ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).

% le_iff_add
thf(fact_2519_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_right_mono
thf(fact_2520_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_2521_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_2522_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_2523_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( B
           != ( plus_plus_nat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_2524_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_left_mono
thf(fact_2525_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_2526_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_2527_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_2528_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_2529_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_2530_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_2531_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_2532_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2533_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2534_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2535_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2536_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2537_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2538_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2539_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2540_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2541_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2542_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2543_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2544_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2545_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_2546_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_2547_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_2548_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_2549_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_2550_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2551_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_2552_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2553_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2554_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2555_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_2556_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_2557_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_2558_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_2559_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2560_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2561_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2562_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2563_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2564_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2565_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2566_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2567_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2568_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2569_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2570_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2571_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2572_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_2573_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_2574_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_2575_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2576_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_2577_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_2578_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_2579_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2580_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_2581_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_2582_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_2583_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2584_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_2585_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_2586_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_2587_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2588_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_2589_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_2590_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_2591_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
       != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_2592_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_2593_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_2594_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_2595_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_2596_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_2597_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_2598_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_2599_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( times_times_real @ A @ B )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_2600_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_2601_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_2602_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_2603_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2604_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_2605_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_2606_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_2607_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2608_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_2609_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_2610_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_2611_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M3: int] :
          ? [N4: int] :
            ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N4 ) )
            & ( member_int @ N4 @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_2612_singletonD,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_2613_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_2614_singletonD,axiom,
    ! [B: set_nat_rat,A: set_nat_rat] :
      ( ( member_set_nat_rat @ B @ ( insert_set_nat_rat @ A @ bot_bo6797373522285170759at_rat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_2615_singletonD,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_2616_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_2617_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_2618_singletonD,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_2619_singleton__iff,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_2620_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_2621_singleton__iff,axiom,
    ! [B: set_nat_rat,A: set_nat_rat] :
      ( ( member_set_nat_rat @ B @ ( insert_set_nat_rat @ A @ bot_bo6797373522285170759at_rat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_2622_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_2623_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_2624_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_2625_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_2626_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2627_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2628_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2629_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2630_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
        = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2631_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A @ A2 )
     != bot_bo2099793752762293965at_nat ) ).

% insert_not_empty
thf(fact_2632_insert__not__empty,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real @ A @ A2 )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_2633_insert__not__empty,axiom,
    ! [A: $o,A2: set_o] :
      ( ( insert_o @ A @ A2 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_2634_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ A2 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_2635_insert__not__empty,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ A2 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_2636_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_2637_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( ( insert_real @ A @ bot_bot_set_real )
        = ( insert_real @ B @ bot_bot_set_real ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_2638_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o @ A @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_2639_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_2640_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( ( insert_int @ A @ bot_bot_set_int )
        = ( insert_int @ B @ bot_bot_set_int ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_2641_finite_OinsertI,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( finite_finite_real @ ( insert_real @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_2642_finite_OinsertI,axiom,
    ! [A2: set_o,A: $o] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_o @ ( insert_o @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_2643_finite_OinsertI,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_2644_finite_OinsertI,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( finite_finite_int @ ( insert_int @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_2645_finite_OinsertI,axiom,
    ! [A2: set_complex,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_2646_finite_OinsertI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_2647_finite_OinsertI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite4001608067531595151d_enat @ ( insert_Extended_enat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_2648_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M3: int] :
          ? [N4: int] :
            ( ( ord_less_int @ M3 @ ( abs_abs_int @ N4 ) )
            & ( member_int @ N4 @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_2649_insert__mono,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ D4 )
     => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ C2 ) @ ( insert8211810215607154385at_nat @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_2650_insert__mono,axiom,
    ! [C2: set_real,D4: set_real,A: real] :
      ( ( ord_less_eq_set_real @ C2 @ D4 )
     => ( ord_less_eq_set_real @ ( insert_real @ A @ C2 ) @ ( insert_real @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_2651_insert__mono,axiom,
    ! [C2: set_o,D4: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ C2 @ D4 )
     => ( ord_less_eq_set_o @ ( insert_o @ A @ C2 ) @ ( insert_o @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_2652_insert__mono,axiom,
    ! [C2: set_nat,D4: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C2 @ D4 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_2653_insert__mono,axiom,
    ! [C2: set_int,D4: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C2 @ D4 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C2 ) @ ( insert_int @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_2654_subset__insert,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
        = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_2655_subset__insert,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
        = ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_2656_subset__insert,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ~ ( member_o @ X @ A2 )
     => ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
        = ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_2657_subset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
        = ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_2658_subset__insert,axiom,
    ! [X: set_nat_rat,A2: set_set_nat_rat,B2: set_set_nat_rat] :
      ( ~ ( member_set_nat_rat @ X @ A2 )
     => ( ( ord_le4375437777232675859at_rat @ A2 @ ( insert_set_nat_rat @ X @ B2 ) )
        = ( ord_le4375437777232675859at_rat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_2659_subset__insert,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_2660_subset__insert,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
        = ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_2661_subset__insertI,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).

% subset_insertI
thf(fact_2662_subset__insertI,axiom,
    ! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A @ B2 ) ) ).

% subset_insertI
thf(fact_2663_subset__insertI,axiom,
    ! [B2: set_o,A: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o @ A @ B2 ) ) ).

% subset_insertI
thf(fact_2664_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).

% subset_insertI
thf(fact_2665_subset__insertI,axiom,
    ! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).

% subset_insertI
thf(fact_2666_subset__insertI2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_2667_subset__insertI2,axiom,
    ! [A2: set_real,B2: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_2668_subset__insertI2,axiom,
    ! [A2: set_o,B2: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A2 @ B2 )
     => ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_2669_subset__insertI2,axiom,
    ! [A2: set_nat,B2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_2670_subset__insertI2,axiom,
    ! [A2: set_int,B2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_2671_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ X5 @ A2 )
       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ X5 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_2672_insert__subsetI,axiom,
    ! [X: real,A2: set_real,X5: set_real] :
      ( ( member_real @ X @ A2 )
     => ( ( ord_less_eq_set_real @ X5 @ A2 )
       => ( ord_less_eq_set_real @ ( insert_real @ X @ X5 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_2673_insert__subsetI,axiom,
    ! [X: $o,A2: set_o,X5: set_o] :
      ( ( member_o @ X @ A2 )
     => ( ( ord_less_eq_set_o @ X5 @ A2 )
       => ( ord_less_eq_set_o @ ( insert_o @ X @ X5 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_2674_insert__subsetI,axiom,
    ! [X: set_nat,A2: set_set_nat,X5: set_set_nat] :
      ( ( member_set_nat @ X @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ X5 @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ X5 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_2675_insert__subsetI,axiom,
    ! [X: set_nat_rat,A2: set_set_nat_rat,X5: set_set_nat_rat] :
      ( ( member_set_nat_rat @ X @ A2 )
     => ( ( ord_le4375437777232675859at_rat @ X5 @ A2 )
       => ( ord_le4375437777232675859at_rat @ ( insert_set_nat_rat @ X @ X5 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_2676_insert__subsetI,axiom,
    ! [X: nat,A2: set_nat,X5: set_nat] :
      ( ( member_nat @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ X5 @ A2 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X @ X5 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_2677_insert__subsetI,axiom,
    ! [X: int,A2: set_int,X5: set_int] :
      ( ( member_int @ X @ A2 )
     => ( ( ord_less_eq_set_int @ X5 @ A2 )
       => ( ord_less_eq_set_int @ ( insert_int @ X @ X5 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_2678_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_2679_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_2680_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_2681_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_2682_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_2683_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_2684_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_2685_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_2686_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2687_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_2688_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_2689_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_2690_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_2691_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_2692_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_2693_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_2694_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_2695_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_2696_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_2697_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_2698_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_2699_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_2700_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_2701_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_2702_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_2703_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_2704_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_2705_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_2706_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_2707_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_2708_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_2709_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_2710_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_2711_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_2712_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_2713_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_2714_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_2715_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2716_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2717_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_2718_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_2719_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_2720_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2721_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_2722_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_2723_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_2724_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2725_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_2726_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_2727_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_2728_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_2729_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_2730_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_2731_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_2732_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_2733_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_2734_Ints__0,axiom,
    member_real @ zero_zero_real @ ring_1_Ints_real ).

% Ints_0
thf(fact_2735_Ints__0,axiom,
    member_rat @ zero_zero_rat @ ring_1_Ints_rat ).

% Ints_0
thf(fact_2736_Ints__0,axiom,
    member_int @ zero_zero_int @ ring_1_Ints_int ).

% Ints_0
thf(fact_2737_Suc__mult__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M2 )
        = ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( M2 = N ) ) ).

% Suc_mult_cancel1
thf(fact_2738_insert__Diff__if,axiom,
    ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ B2 )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
          = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ B2 )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
          = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2739_insert__Diff__if,axiom,
    ! [X: real,B2: set_real,A2: set_real] :
      ( ( ( member_real @ X @ B2 )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
          = ( minus_minus_set_real @ A2 @ B2 ) ) )
      & ( ~ ( member_real @ X @ B2 )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
          = ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2740_insert__Diff__if,axiom,
    ! [X: $o,B2: set_o,A2: set_o] :
      ( ( ( member_o @ X @ B2 )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
          = ( minus_minus_set_o @ A2 @ B2 ) ) )
      & ( ~ ( member_o @ X @ B2 )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
          = ( insert_o @ X @ ( minus_minus_set_o @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2741_insert__Diff__if,axiom,
    ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( ( member_set_nat @ X @ B2 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
      & ( ~ ( member_set_nat @ X @ B2 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
          = ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2742_insert__Diff__if,axiom,
    ! [X: set_nat_rat,B2: set_set_nat_rat,A2: set_set_nat_rat] :
      ( ( ( member_set_nat_rat @ X @ B2 )
       => ( ( minus_1626877696091177228at_rat @ ( insert_set_nat_rat @ X @ A2 ) @ B2 )
          = ( minus_1626877696091177228at_rat @ A2 @ B2 ) ) )
      & ( ~ ( member_set_nat_rat @ X @ B2 )
       => ( ( minus_1626877696091177228at_rat @ ( insert_set_nat_rat @ X @ A2 ) @ B2 )
          = ( insert_set_nat_rat @ X @ ( minus_1626877696091177228at_rat @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2743_insert__Diff__if,axiom,
    ! [X: int,B2: set_int,A2: set_int] :
      ( ( ( member_int @ X @ B2 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
          = ( minus_minus_set_int @ A2 @ B2 ) ) )
      & ( ~ ( member_int @ X @ B2 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
          = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2744_insert__Diff__if,axiom,
    ! [X: nat,B2: set_nat,A2: set_nat] :
      ( ( ( member_nat @ X @ B2 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
          = ( minus_minus_set_nat @ A2 @ B2 ) ) )
      & ( ~ ( member_nat @ X @ B2 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
          = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2745_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2746_Ints__1,axiom,
    member_complex @ one_one_complex @ ring_1_Ints_complex ).

% Ints_1
thf(fact_2747_Ints__1,axiom,
    member_real @ one_one_real @ ring_1_Ints_real ).

% Ints_1
thf(fact_2748_Ints__1,axiom,
    member_rat @ one_one_rat @ ring_1_Ints_rat ).

% Ints_1
thf(fact_2749_Ints__1,axiom,
    member_int @ one_one_int @ ring_1_Ints_int ).

% Ints_1
thf(fact_2750_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_2751_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_2752_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_2753_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_2754_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_2755_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_2756_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_2757_le__square,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).

% le_square
thf(fact_2758_le__cube,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).

% le_cube
thf(fact_2759_diff__mult__distrib2,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_2760_diff__mult__distrib,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_2761_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_2762_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_2763_abs__add__one__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_2764_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_2765_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_2766_Ints__nonzero__abs__ge1,axiom,
    ! [X: real] :
      ( ( member_real @ X @ ring_1_Ints_real )
     => ( ( X != zero_zero_real )
       => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_2767_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_2768_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_2769_Ints__nonzero__abs__less1,axiom,
    ! [X: real] :
      ( ( member_real @ X @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
       => ( X = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_2770_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_2771_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_2772_Ints__eq__abs__less1,axiom,
    ! [X: real,Y: real] :
      ( ( member_real @ X @ ring_1_Ints_real )
     => ( ( member_real @ Y @ ring_1_Ints_real )
       => ( ( X = Y )
          = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_2773_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( member_rat @ Y @ ring_1_Ints_rat )
       => ( ( X = Y )
          = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y ) ) @ one_one_rat ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_2774_Ints__eq__abs__less1,axiom,
    ! [X: int,Y: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( member_int @ Y @ ring_1_Ints_int )
       => ( ( X = Y )
          = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_2775_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2776_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ Y @ 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 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2777_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ 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 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2778_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
        = ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% Ints_odd_less_0
thf(fact_2779_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_2780_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_2781_subset__decode__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M2 ) @ ( nat_set_decode @ N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% subset_decode_imp_le
thf(fact_2782_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_real @ X @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2783_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ( ord_less_rat @ Y @ 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 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2784_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ 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 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2785_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_2786_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_2787_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_2788_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_2789_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_2790_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_2791_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_2792_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_2793_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_2794_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( ord_less_eq_real @ R2 @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2795_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2796_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_2797_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_2798_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_2799_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_2800_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_2801_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_2802_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_2803_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_2804_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_2805_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_2806_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_2807_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_2808_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( suc @ ( nat2 @ Z ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_2809_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_2810_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_2811_finite__set__decode,axiom,
    ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).

% finite_set_decode
thf(fact_2812_ex__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X6: nat] : ( P2 @ X6 ) )
    = ( ^ [P3: nat > $o] :
        ? [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
          & ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).

% ex_nat
thf(fact_2813_all__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ! [X6: nat] : ( P2 @ X6 ) )
    = ( ^ [P3: nat > $o] :
        ! [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).

% all_nat
thf(fact_2814_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z6 ) )
          = ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_2815_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2816_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2817_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2818_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2819_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2820_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2821_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2822_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2823_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_2824_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_2825_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_2826_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_2827_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2828_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_2829_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_2830_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_2831_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2832_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_2833_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_2834_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_2835_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2836_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_2837_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_2838_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_2839_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2840_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_2841_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_2842_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_2843_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2844_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_2845_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_2846_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_2847_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2848_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_2849_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_2850_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_2851_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2852_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_2853_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_2854_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_2855_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2856_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2857_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2858_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2859_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2860_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2861_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2862_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2863_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_2864_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_2865_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_2866_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_2867_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_2868_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_2869_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_2870_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_2871_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_2872_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2873_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_2874_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_2875_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_2876_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_2877_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_2878_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_2879_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_2880_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2881_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_2882_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_2883_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_2884_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2885_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_2886_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_2887_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_2888_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2889_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2890_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2891_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2892_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_2893_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_2894_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2895_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_2896_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_2897_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2898_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_2899_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_2900_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2901_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_2902_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_2903_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_2904_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2905_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_2906_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_2907_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2908_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_2909_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_2910_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_2911_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2912_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_2913_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_2914_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_2915_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_2916_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_2917_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_2918_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2919_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_2920_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_2921_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_2922_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2923_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_2924_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_2925_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_2926_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2927_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_2928_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_2929_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_2930_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2931_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_2932_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_2933_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_2934_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2935_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_2936_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_2937_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2938_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_2939_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_2940_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_2941_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_2942_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_2943_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_2944_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_2945_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_2946_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_2947_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_2948_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_2949_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_2950_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_2951_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_2952_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_2953_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_2954_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_2955_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_2956_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_2957_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2958_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2959_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2960_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2961_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_2962_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_2963_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_2964_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_2965_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_2966_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_2967_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_2968_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_2969_add__mono1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).

% add_mono1
thf(fact_2970_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_2971_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_2972_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_2973_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2974_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_2975_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_2976_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2977_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2978_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2979_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2980_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_2981_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_2982_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_2983_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_2984_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_2985_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_2986_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_2987_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_2988_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_2989_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_2990_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2991_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_2992_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_2993_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_2994_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_2995_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_2996_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_2997_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_2998_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2999_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_3000_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_3001_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_3002_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_3003_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_3004_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_3005_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_3006_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_3007_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_3008_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_3009_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_3010_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_3011_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_3012_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_3013_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_3014_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_3015_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_3016_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_3017_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_3018_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3019_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_3020_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_3021_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_3022_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_3023_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_3024_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_3025_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_3026_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_3027_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_3028_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3029_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_3030_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_3031_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_3032_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_3033_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_3034_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_3035_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3036_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_3037_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_3038_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_3039_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3040_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_3041_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_3042_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3043_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_3044_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_3045_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ ( ord_less_real @ A @ B )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3046_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_3047_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_3048_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_3049_infinite__finite__induct,axiom,
    ! [P: set_set_nat > $o,A2: set_set_nat] :
      ( ! [A3: set_set_nat] :
          ( ~ ( finite1152437895449049373et_nat @ A3 )
         => ( P @ A3 ) )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X4: set_nat,F3: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F3 )
             => ( ~ ( member_set_nat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_set_nat @ X4 @ F3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3050_infinite__finite__induct,axiom,
    ! [P: set_set_nat_rat > $o,A2: set_set_nat_rat] :
      ( ! [A3: set_set_nat_rat] :
          ( ~ ( finite6430367030675640852at_rat @ A3 )
         => ( P @ A3 ) )
     => ( ( P @ bot_bo6797373522285170759at_rat )
       => ( ! [X4: set_nat_rat,F3: set_set_nat_rat] :
              ( ( finite6430367030675640852at_rat @ F3 )
             => ( ~ ( member_set_nat_rat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_set_nat_rat @ X4 @ F3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3051_infinite__finite__induct,axiom,
    ! [P: set_complex > $o,A2: set_complex] :
      ( ! [A3: set_complex] :
          ( ~ ( finite3207457112153483333omplex @ A3 )
         => ( P @ A3 ) )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,F3: set_complex] :
              ( ( finite3207457112153483333omplex @ F3 )
             => ( ~ ( member_complex @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_complex @ X4 @ F3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3052_infinite__finite__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat] :
          ( ~ ( finite6177210948735845034at_nat @ A3 )
         => ( P @ A3 ) )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F3 )
             => ( ~ ( member8440522571783428010at_nat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert8211810215607154385at_nat @ X4 @ F3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3053_infinite__finite__induct,axiom,
    ! [P: set_Extended_enat > $o,A2: set_Extended_enat] :
      ( ! [A3: set_Extended_enat] :
          ( ~ ( finite4001608067531595151d_enat @ A3 )
         => ( P @ A3 ) )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat,F3: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ F3 )
             => ( ~ ( member_Extended_enat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_Extended_enat @ X4 @ F3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3054_infinite__finite__induct,axiom,
    ! [P: set_real > $o,A2: set_real] :
      ( ! [A3: set_real] :
          ( ~ ( finite_finite_real @ A3 )
         => ( P @ A3 ) )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,F3: set_real] :
              ( ( finite_finite_real @ F3 )
             => ( ~ ( member_real @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_real @ X4 @ F3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3055_infinite__finite__induct,axiom,
    ! [P: set_o > $o,A2: set_o] :
      ( ! [A3: set_o] :
          ( ~ ( finite_finite_o @ A3 )
         => ( P @ A3 ) )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X4: $o,F3: set_o] :
              ( ( finite_finite_o @ F3 )
             => ( ~ ( member_o @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_o @ X4 @ F3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3056_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A2: set_nat] :
      ( ! [A3: set_nat] :
          ( ~ ( finite_finite_nat @ A3 )
         => ( P @ A3 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3057_infinite__finite__induct,axiom,
    ! [P: set_int > $o,A2: set_int] :
      ( ! [A3: set_int] :
          ( ~ ( finite_finite_int @ A3 )
         => ( P @ A3 ) )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X4: int,F3: set_int] :
              ( ( finite_finite_int @ F3 )
             => ( ~ ( member_int @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_int @ X4 @ F3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3058_finite__ne__induct,axiom,
    ! [F2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( F2 != bot_bot_set_set_nat )
       => ( ! [X4: set_nat] : ( P @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
         => ( ! [X4: set_nat,F3: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F3 )
               => ( ( F3 != bot_bot_set_set_nat )
                 => ( ~ ( member_set_nat @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_set_nat @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3059_finite__ne__induct,axiom,
    ! [F2: set_set_nat_rat,P: set_set_nat_rat > $o] :
      ( ( finite6430367030675640852at_rat @ F2 )
     => ( ( F2 != bot_bo6797373522285170759at_rat )
       => ( ! [X4: set_nat_rat] : ( P @ ( insert_set_nat_rat @ X4 @ bot_bo6797373522285170759at_rat ) )
         => ( ! [X4: set_nat_rat,F3: set_set_nat_rat] :
                ( ( finite6430367030675640852at_rat @ F3 )
               => ( ( F3 != bot_bo6797373522285170759at_rat )
                 => ( ~ ( member_set_nat_rat @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_set_nat_rat @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3060_finite__ne__induct,axiom,
    ! [F2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( F2 != bot_bot_set_complex )
       => ( ! [X4: complex] : ( P @ ( insert_complex @ X4 @ bot_bot_set_complex ) )
         => ( ! [X4: complex,F3: set_complex] :
                ( ( finite3207457112153483333omplex @ F3 )
               => ( ( F3 != bot_bot_set_complex )
                 => ( ~ ( member_complex @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_complex @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3061_finite__ne__induct,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( F2 != bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) )
         => ( ! [X4: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F3 )
               => ( ( F3 != bot_bo2099793752762293965at_nat )
                 => ( ~ ( member8440522571783428010at_nat @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert8211810215607154385at_nat @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3062_finite__ne__induct,axiom,
    ! [F2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( F2 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] : ( P @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) )
         => ( ! [X4: extended_enat,F3: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F3 )
               => ( ( F3 != bot_bo7653980558646680370d_enat )
                 => ( ~ ( member_Extended_enat @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_Extended_enat @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3063_finite__ne__induct,axiom,
    ! [F2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( F2 != bot_bot_set_real )
       => ( ! [X4: real] : ( P @ ( insert_real @ X4 @ bot_bot_set_real ) )
         => ( ! [X4: real,F3: set_real] :
                ( ( finite_finite_real @ F3 )
               => ( ( F3 != bot_bot_set_real )
                 => ( ~ ( member_real @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_real @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3064_finite__ne__induct,axiom,
    ! [F2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( F2 != bot_bot_set_o )
       => ( ! [X4: $o] : ( P @ ( insert_o @ X4 @ bot_bot_set_o ) )
         => ( ! [X4: $o,F3: set_o] :
                ( ( finite_finite_o @ F3 )
               => ( ( F3 != bot_bot_set_o )
                 => ( ~ ( member_o @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_o @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3065_finite__ne__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
         => ( ! [X4: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3066_finite__ne__induct,axiom,
    ! [F2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( F2 != bot_bot_set_int )
       => ( ! [X4: int] : ( P @ ( insert_int @ X4 @ bot_bot_set_int ) )
         => ( ! [X4: int,F3: set_int] :
                ( ( finite_finite_int @ F3 )
               => ( ( F3 != bot_bot_set_int )
                 => ( ~ ( member_int @ X4 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_int @ X4 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3067_finite__induct,axiom,
    ! [F2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X4: set_nat,F3: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F3 )
             => ( ~ ( member_set_nat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_set_nat @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_3068_finite__induct,axiom,
    ! [F2: set_set_nat_rat,P: set_set_nat_rat > $o] :
      ( ( finite6430367030675640852at_rat @ F2 )
     => ( ( P @ bot_bo6797373522285170759at_rat )
       => ( ! [X4: set_nat_rat,F3: set_set_nat_rat] :
              ( ( finite6430367030675640852at_rat @ F3 )
             => ( ~ ( member_set_nat_rat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_set_nat_rat @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_3069_finite__induct,axiom,
    ! [F2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,F3: set_complex] :
              ( ( finite3207457112153483333omplex @ F3 )
             => ( ~ ( member_complex @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_complex @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_3070_finite__induct,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F3 )
             => ( ~ ( member8440522571783428010at_nat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert8211810215607154385at_nat @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_3071_finite__induct,axiom,
    ! [F2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat,F3: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ F3 )
             => ( ~ ( member_Extended_enat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_Extended_enat @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_3072_finite__induct,axiom,
    ! [F2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,F3: set_real] :
              ( ( finite_finite_real @ F3 )
             => ( ~ ( member_real @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_real @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_3073_finite__induct,axiom,
    ! [F2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X4: $o,F3: set_o] :
              ( ( finite_finite_o @ F3 )
             => ( ~ ( member_o @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_o @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_3074_finite__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_3075_finite__induct,axiom,
    ! [F2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X4: int,F3: set_int] :
              ( ( finite_finite_int @ F3 )
             => ( ~ ( member_int @ X4 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_int @ X4 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_3076_finite_Osimps,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A4: set_complex] :
          ( ( A4 = bot_bot_set_complex )
          | ? [A6: set_complex,B4: complex] :
              ( ( A4
                = ( insert_complex @ B4 @ A6 ) )
              & ( finite3207457112153483333omplex @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_3077_finite_Osimps,axiom,
    ( finite6177210948735845034at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat] :
          ( ( A4 = bot_bo2099793752762293965at_nat )
          | ? [A6: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat] :
              ( ( A4
                = ( insert8211810215607154385at_nat @ B4 @ A6 ) )
              & ( finite6177210948735845034at_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_3078_finite_Osimps,axiom,
    ( finite4001608067531595151d_enat
    = ( ^ [A4: set_Extended_enat] :
          ( ( A4 = bot_bo7653980558646680370d_enat )
          | ? [A6: set_Extended_enat,B4: extended_enat] :
              ( ( A4
                = ( insert_Extended_enat @ B4 @ A6 ) )
              & ( finite4001608067531595151d_enat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_3079_finite_Osimps,axiom,
    ( finite_finite_real
    = ( ^ [A4: set_real] :
          ( ( A4 = bot_bot_set_real )
          | ? [A6: set_real,B4: real] :
              ( ( A4
                = ( insert_real @ B4 @ A6 ) )
              & ( finite_finite_real @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_3080_finite_Osimps,axiom,
    ( finite_finite_o
    = ( ^ [A4: set_o] :
          ( ( A4 = bot_bot_set_o )
          | ? [A6: set_o,B4: $o] :
              ( ( A4
                = ( insert_o @ B4 @ A6 ) )
              & ( finite_finite_o @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_3081_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A4: set_nat] :
          ( ( A4 = bot_bot_set_nat )
          | ? [A6: set_nat,B4: nat] :
              ( ( A4
                = ( insert_nat @ B4 @ A6 ) )
              & ( finite_finite_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_3082_finite_Osimps,axiom,
    ( finite_finite_int
    = ( ^ [A4: set_int] :
          ( ( A4 = bot_bot_set_int )
          | ? [A6: set_int,B4: int] :
              ( ( A4
                = ( insert_int @ B4 @ A6 ) )
              & ( finite_finite_int @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_3083_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A3: set_complex] :
              ( ? [A5: complex] :
                  ( A
                  = ( insert_complex @ A5 @ A3 ) )
             => ~ ( finite3207457112153483333omplex @ A3 ) ) ) ) ).

% finite.cases
thf(fact_3084_finite_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ~ ! [A3: set_Pr1261947904930325089at_nat] :
              ( ? [A5: product_prod_nat_nat] :
                  ( A
                  = ( insert8211810215607154385at_nat @ A5 @ A3 ) )
             => ~ ( finite6177210948735845034at_nat @ A3 ) ) ) ) ).

% finite.cases
thf(fact_3085_finite_Ocases,axiom,
    ! [A: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A )
     => ( ( A != bot_bo7653980558646680370d_enat )
       => ~ ! [A3: set_Extended_enat] :
              ( ? [A5: extended_enat] :
                  ( A
                  = ( insert_Extended_enat @ A5 @ A3 ) )
             => ~ ( finite4001608067531595151d_enat @ A3 ) ) ) ) ).

% finite.cases
thf(fact_3086_finite_Ocases,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ~ ! [A3: set_real] :
              ( ? [A5: real] :
                  ( A
                  = ( insert_real @ A5 @ A3 ) )
             => ~ ( finite_finite_real @ A3 ) ) ) ) ).

% finite.cases
thf(fact_3087_finite_Ocases,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ~ ! [A3: set_o] :
              ( ? [A5: $o] :
                  ( A
                  = ( insert_o @ A5 @ A3 ) )
             => ~ ( finite_finite_o @ A3 ) ) ) ) ).

% finite.cases
thf(fact_3088_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A3: set_nat] :
              ( ? [A5: nat] :
                  ( A
                  = ( insert_nat @ A5 @ A3 ) )
             => ~ ( finite_finite_nat @ A3 ) ) ) ) ).

% finite.cases
thf(fact_3089_finite_Ocases,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ~ ! [A3: set_int] :
              ( ? [A5: int] :
                  ( A
                  = ( insert_int @ A5 @ A3 ) )
             => ~ ( finite_finite_int @ A3 ) ) ) ) ).

% finite.cases
thf(fact_3090_subset__singleton__iff,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ X5 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( ( X5 = bot_bo2099793752762293965at_nat )
        | ( X5
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_3091_subset__singleton__iff,axiom,
    ! [X5: set_real,A: real] :
      ( ( ord_less_eq_set_real @ X5 @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( ( X5 = bot_bot_set_real )
        | ( X5
          = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_3092_subset__singleton__iff,axiom,
    ! [X5: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ X5 @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( ( X5 = bot_bot_set_o )
        | ( X5
          = ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_3093_subset__singleton__iff,axiom,
    ! [X5: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X5 = bot_bot_set_nat )
        | ( X5
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_3094_subset__singleton__iff,axiom,
    ! [X5: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X5 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X5 = bot_bot_set_int )
        | ( X5
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_3095_subset__singletonD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
     => ( ( A2 = bot_bo2099793752762293965at_nat )
        | ( A2
          = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_3096_subset__singletonD,axiom,
    ! [A2: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) )
     => ( ( A2 = bot_bot_set_real )
        | ( A2
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_3097_subset__singletonD,axiom,
    ! [A2: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) )
     => ( ( A2 = bot_bot_set_o )
        | ( A2
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_3098_subset__singletonD,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A2 = bot_bot_set_nat )
        | ( A2
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_3099_subset__singletonD,axiom,
    ! [A2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
     => ( ( A2 = bot_bot_set_int )
        | ( A2
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_3100_less__1__mult,axiom,
    ! [M2: real,N: real] :
      ( ( ord_less_real @ one_one_real @ M2 )
     => ( ( ord_less_real @ one_one_real @ N )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_3101_less__1__mult,axiom,
    ! [M2: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M2 )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_3102_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M2 )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_3103_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M2 )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_3104_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W2 @ Z ) )
          = ( ( times_times_rat @ X @ Z )
            = ( times_times_rat @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3105_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W2 @ Z ) )
          = ( ( times_times_real @ X @ Z )
            = ( times_times_real @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3106_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_3107_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_3108_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_3109_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_3110_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_3111_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_3112_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_3113_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B )
       => ( A
          = ( divide_divide_real @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_3114_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_3115_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B @ C )
          = A )
        = ( B
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_3116_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_3117_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B @ C ) )
        = ( ( times_times_real @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_3118_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_3119_Diff__insert__absorb,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_3120_Diff__insert__absorb,axiom,
    ! [X: set_nat_rat,A2: set_set_nat_rat] :
      ( ~ ( member_set_nat_rat @ X @ A2 )
     => ( ( minus_1626877696091177228at_rat @ ( insert_set_nat_rat @ X @ A2 ) @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_3121_Diff__insert__absorb,axiom,
    ! [X: real,A2: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_3122_Diff__insert__absorb,axiom,
    ! [X: $o,A2: set_o] :
      ( ~ ( member_o @ X @ A2 )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_3123_Diff__insert__absorb,axiom,
    ! [X: int,A2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_3124_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_3125_Diff__insert2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_3126_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_3127_Diff__insert2,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      ( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_3128_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_3129_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_3130_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_3131_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ A2 )
     => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_3132_insert__Diff,axiom,
    ! [A: set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ A @ A2 )
     => ( ( insert_set_nat_rat @ A @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ A @ bot_bo6797373522285170759at_rat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_3133_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_3134_insert__Diff,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_3135_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_3136_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_3137_Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).

% Diff_insert
thf(fact_3138_Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_3139_Diff__insert,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      ( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B2 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_3140_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_3141_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_3142_subset__Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ ( insert8211810215607154385at_nat @ X @ C2 ) ) )
      = ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ C2 ) )
        & ~ ( member8440522571783428010at_nat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_3143_subset__Diff__insert,axiom,
    ! [A2: set_real,B2: set_real,X: real,C2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C2 ) )
        & ~ ( member_real @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_3144_subset__Diff__insert,axiom,
    ! [A2: set_o,B2: set_o,X: $o,C2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ ( insert_o @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ C2 ) )
        & ~ ( member_o @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_3145_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X @ C2 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C2 ) )
        & ~ ( member_set_nat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_3146_subset__Diff__insert,axiom,
    ! [A2: set_set_nat_rat,B2: set_set_nat_rat,X: set_nat_rat,C2: set_set_nat_rat] :
      ( ( ord_le4375437777232675859at_rat @ A2 @ ( minus_1626877696091177228at_rat @ B2 @ ( insert_set_nat_rat @ X @ C2 ) ) )
      = ( ( ord_le4375437777232675859at_rat @ A2 @ ( minus_1626877696091177228at_rat @ B2 @ C2 ) )
        & ~ ( member_set_nat_rat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_3147_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
        & ~ ( member_nat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_3148_subset__Diff__insert,axiom,
    ! [A2: set_int,B2: set_int,X: int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C2 ) )
        & ~ ( member_int @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_3149_card__insert__le,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A2 ) @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_3150_card__insert__le,axiom,
    ! [A2: set_real,X: real] : ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ ( insert_real @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_3151_card__insert__le,axiom,
    ! [A2: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ ( finite_card_o @ ( insert_o @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_3152_card__insert__le,axiom,
    ! [A2: set_complex,X: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ ( insert_complex @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_3153_card__insert__le,axiom,
    ! [A2: set_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ ( insert_list_nat @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_3154_card__insert__le,axiom,
    ! [A2: set_set_nat,X: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_3155_card__insert__le,axiom,
    ! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_3156_card__insert__le,axiom,
    ! [A2: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ ( insert_int @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_3157_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_3158_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_3159_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_3160_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_3161_less__mult__imp__div__less,axiom,
    ! [M2: nat,I: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_3162_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2
        = ( times_times_nat @ M2 @ N ) )
     => ( ( N = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_3163_div__times__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).

% div_times_less_eq_dividend
thf(fact_3164_times__div__less__eq__dividend,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).

% times_div_less_eq_dividend
thf(fact_3165_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_3166_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z2: int] :
        ? [N4: nat] :
          ( Z2
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_3167_frac__1__eq,axiom,
    ! [X: real] :
      ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( archim2898591450579166408c_real @ X ) ) ).

% frac_1_eq
thf(fact_3168_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_3169_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X3: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_3170_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_3171_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_3172_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_3173_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_3174_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_3175_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
        = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_3176_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
        = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_3177_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W2 @ Z ) ) ) ).

% nat_mono_iff
thf(fact_3178_zless__nat__eq__int__zless,axiom,
    ! [M2: nat,Z: int] :
      ( ( ord_less_nat @ M2 @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_3179_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_3180_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_3181_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_3182_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_3183_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_3184_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_3185_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3186_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_3187_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_3188_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_3189_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_3190_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_3191_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_3192_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_3193_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3194_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_3195_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_3196_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_3197_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3198_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_3199_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_3200_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_3201_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3202_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_3203_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_3204_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_3205_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_3206_int__eq__iff,axiom,
    ! [M2: nat,Z: int] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = Z )
      = ( ( M2
          = ( nat2 @ Z ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).

% int_eq_iff
thf(fact_3207_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_3208_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X4 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3209_finite__ranking__induct,axiom,
    ! [S2: set_Extended_enat,P: set_Extended_enat > $o,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat,S4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ S4 )
             => ( ! [Y4: extended_enat] :
                    ( ( member_Extended_enat @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_Extended_enat @ X4 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3210_finite__ranking__induct,axiom,
    ! [S2: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y4: real] :
                    ( ( member_real @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X4 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3211_finite__ranking__induct,axiom,
    ! [S2: set_o,P: set_o > $o,F: $o > rat] :
      ( ( finite_finite_o @ S2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X4: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y4: $o] :
                    ( ( member_o @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X4 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3212_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y4: nat] :
                    ( ( member_nat @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X4 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3213_finite__ranking__induct,axiom,
    ! [S2: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X4: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y4: int] :
                    ( ( member_int @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X4 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3214_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X4 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3215_finite__ranking__induct,axiom,
    ! [S2: set_Extended_enat,P: set_Extended_enat > $o,F: extended_enat > num] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat,S4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ S4 )
             => ( ! [Y4: extended_enat] :
                    ( ( member_Extended_enat @ Y4 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_Extended_enat @ X4 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3216_finite__ranking__induct,axiom,
    ! [S2: set_real,P: set_real > $o,F: real > num] :
      ( ( finite_finite_real @ S2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y4: real] :
                    ( ( member_real @ Y4 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X4 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3217_finite__ranking__induct,axiom,
    ! [S2: set_o,P: set_o > $o,F: $o > num] :
      ( ( finite_finite_o @ S2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X4: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y4: $o] :
                    ( ( member_o @ Y4 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X4 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3218_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3219_discrete,axiom,
    ( ord_less_int
    = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3220_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_3221_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3222_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3223_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3224_finite__linorder__max__induct,axiom,
    ! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [B5: extended_enat,A3: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A3 )
             => ( ! [X2: extended_enat] :
                    ( ( member_Extended_enat @ X2 @ A3 )
                   => ( ord_le72135733267957522d_enat @ X2 @ B5 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_Extended_enat @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3225_finite__linorder__max__induct,axiom,
    ! [A2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B5: $o,A3: set_o] :
              ( ( finite_finite_o @ A3 )
             => ( ! [X2: $o] :
                    ( ( member_o @ X2 @ A3 )
                   => ( ord_less_o @ X2 @ B5 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_o @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3226_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B5: real,A3: set_real] :
              ( ( finite_finite_real @ A3 )
             => ( ! [X2: real] :
                    ( ( member_real @ X2 @ A3 )
                   => ( ord_less_real @ X2 @ B5 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_real @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3227_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B5: rat,A3: set_rat] :
              ( ( finite_finite_rat @ A3 )
             => ( ! [X2: rat] :
                    ( ( member_rat @ X2 @ A3 )
                   => ( ord_less_rat @ X2 @ B5 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_rat @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3228_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B5: num,A3: set_num] :
              ( ( finite_finite_num @ A3 )
             => ( ! [X2: num] :
                    ( ( member_num @ X2 @ A3 )
                   => ( ord_less_num @ X2 @ B5 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_num @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3229_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A3: set_nat] :
              ( ( finite_finite_nat @ A3 )
             => ( ! [X2: nat] :
                    ( ( member_nat @ X2 @ A3 )
                   => ( ord_less_nat @ X2 @ B5 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_nat @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3230_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A3: set_int] :
              ( ( finite_finite_int @ A3 )
             => ( ! [X2: int] :
                    ( ( member_int @ X2 @ A3 )
                   => ( ord_less_int @ X2 @ B5 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_int @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3231_finite__linorder__min__induct,axiom,
    ! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [B5: extended_enat,A3: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A3 )
             => ( ! [X2: extended_enat] :
                    ( ( member_Extended_enat @ X2 @ A3 )
                   => ( ord_le72135733267957522d_enat @ B5 @ X2 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_Extended_enat @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3232_finite__linorder__min__induct,axiom,
    ! [A2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B5: $o,A3: set_o] :
              ( ( finite_finite_o @ A3 )
             => ( ! [X2: $o] :
                    ( ( member_o @ X2 @ A3 )
                   => ( ord_less_o @ B5 @ X2 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_o @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3233_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B5: real,A3: set_real] :
              ( ( finite_finite_real @ A3 )
             => ( ! [X2: real] :
                    ( ( member_real @ X2 @ A3 )
                   => ( ord_less_real @ B5 @ X2 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_real @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3234_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B5: rat,A3: set_rat] :
              ( ( finite_finite_rat @ A3 )
             => ( ! [X2: rat] :
                    ( ( member_rat @ X2 @ A3 )
                   => ( ord_less_rat @ B5 @ X2 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_rat @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3235_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B5: num,A3: set_num] :
              ( ( finite_finite_num @ A3 )
             => ( ! [X2: num] :
                    ( ( member_num @ X2 @ A3 )
                   => ( ord_less_num @ B5 @ X2 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_num @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3236_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A3: set_nat] :
              ( ( finite_finite_nat @ A3 )
             => ( ! [X2: nat] :
                    ( ( member_nat @ X2 @ A3 )
                   => ( ord_less_nat @ B5 @ X2 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_nat @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3237_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A3: set_int] :
              ( ( finite_finite_int @ A3 )
             => ( ! [X2: int] :
                    ( ( member_int @ X2 @ A3 )
                   => ( ord_less_int @ B5 @ X2 ) )
               => ( ( P @ A3 )
                 => ( P @ ( insert_int @ B5 @ A3 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3238_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3239_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_3240_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_3241_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3242_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_3243_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_3244_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3245_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_3246_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_3247_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_3248_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3249_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_3250_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_3251_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_3252_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3253_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_3254_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_3255_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3256_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_3257_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_3258_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_3259_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3260_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_3261_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_3262_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_3263_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3264_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_3265_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_3266_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3267_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_3268_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_3269_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3270_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_3271_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_3272_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3273_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_3274_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_3275_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_3276_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3277_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_3278_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_3279_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_3280_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3281_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_3282_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_3283_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_3284_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3285_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_3286_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_3287_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_3288_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_3289_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_3290_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_3291_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_3292_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_3293_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).

% gt_half_sum
thf(fact_3294_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_3295_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_3296_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3297_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3298_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3299_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3300_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3301_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3302_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ B @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_3303_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_3304_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_3305_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_3306_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_3307_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_3308_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_3309_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_3310_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_3311_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_3312_finite__subset__induct,axiom,
    ! [F2: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( ord_le6893508408891458716et_nat @ F2 @ A2 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A5: set_nat,F3: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F3 )
               => ( ( member_set_nat @ A5 @ A2 )
                 => ( ~ ( member_set_nat @ A5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_set_nat @ A5 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3313_finite__subset__induct,axiom,
    ! [F2: set_set_nat_rat,A2: set_set_nat_rat,P: set_set_nat_rat > $o] :
      ( ( finite6430367030675640852at_rat @ F2 )
     => ( ( ord_le4375437777232675859at_rat @ F2 @ A2 )
       => ( ( P @ bot_bo6797373522285170759at_rat )
         => ( ! [A5: set_nat_rat,F3: set_set_nat_rat] :
                ( ( finite6430367030675640852at_rat @ F3 )
               => ( ( member_set_nat_rat @ A5 @ A2 )
                 => ( ~ ( member_set_nat_rat @ A5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_set_nat_rat @ A5 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3314_finite__subset__induct,axiom,
    ! [F2: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( ord_le211207098394363844omplex @ F2 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A5: complex,F3: set_complex] :
                ( ( finite3207457112153483333omplex @ F3 )
               => ( ( member_complex @ A5 @ A2 )
                 => ( ~ ( member_complex @ A5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_complex @ A5 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3315_finite__subset__induct,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( ord_le3146513528884898305at_nat @ F2 @ A2 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A5: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F3 )
               => ( ( member8440522571783428010at_nat @ A5 @ A2 )
                 => ( ~ ( member8440522571783428010at_nat @ A5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert8211810215607154385at_nat @ A5 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3316_finite__subset__induct,axiom,
    ! [F2: set_Extended_enat,A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( ord_le7203529160286727270d_enat @ F2 @ A2 )
       => ( ( P @ bot_bo7653980558646680370d_enat )
         => ( ! [A5: extended_enat,F3: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F3 )
               => ( ( member_Extended_enat @ A5 @ A2 )
                 => ( ~ ( member_Extended_enat @ A5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_Extended_enat @ A5 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3317_finite__subset__induct,axiom,
    ! [F2: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( ord_less_eq_set_real @ F2 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A5: real,F3: set_real] :
                ( ( finite_finite_real @ F3 )
               => ( ( member_real @ A5 @ A2 )
                 => ( ~ ( member_real @ A5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_real @ A5 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3318_finite__subset__induct,axiom,
    ! [F2: set_o,A2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( ord_less_eq_set_o @ F2 @ A2 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A5: $o,F3: set_o] :
                ( ( finite_finite_o @ F3 )
               => ( ( member_o @ A5 @ A2 )
                 => ( ~ ( member_o @ A5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_o @ A5 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3319_finite__subset__induct,axiom,
    ! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A5 @ A2 )
                 => ( ~ ( member_nat @ A5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3320_finite__subset__induct,axiom,
    ! [F2: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( ord_less_eq_set_int @ F2 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A5: int,F3: set_int] :
                ( ( finite_finite_int @ F3 )
               => ( ( member_int @ A5 @ A2 )
                 => ( ~ ( member_int @ A5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_int @ A5 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3321_finite__subset__induct_H,axiom,
    ! [F2: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( ord_le6893508408891458716et_nat @ F2 @ A2 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A5: set_nat,F3: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F3 )
               => ( ( member_set_nat @ A5 @ A2 )
                 => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
                   => ( ~ ( member_set_nat @ A5 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_set_nat @ A5 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3322_finite__subset__induct_H,axiom,
    ! [F2: set_set_nat_rat,A2: set_set_nat_rat,P: set_set_nat_rat > $o] :
      ( ( finite6430367030675640852at_rat @ F2 )
     => ( ( ord_le4375437777232675859at_rat @ F2 @ A2 )
       => ( ( P @ bot_bo6797373522285170759at_rat )
         => ( ! [A5: set_nat_rat,F3: set_set_nat_rat] :
                ( ( finite6430367030675640852at_rat @ F3 )
               => ( ( member_set_nat_rat @ A5 @ A2 )
                 => ( ( ord_le4375437777232675859at_rat @ F3 @ A2 )
                   => ( ~ ( member_set_nat_rat @ A5 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_set_nat_rat @ A5 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3323_finite__subset__induct_H,axiom,
    ! [F2: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( ord_le211207098394363844omplex @ F2 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A5: complex,F3: set_complex] :
                ( ( finite3207457112153483333omplex @ F3 )
               => ( ( member_complex @ A5 @ A2 )
                 => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
                   => ( ~ ( member_complex @ A5 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_complex @ A5 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3324_finite__subset__induct_H,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( ord_le3146513528884898305at_nat @ F2 @ A2 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A5: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F3 )
               => ( ( member8440522571783428010at_nat @ A5 @ A2 )
                 => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
                   => ( ~ ( member8440522571783428010at_nat @ A5 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert8211810215607154385at_nat @ A5 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3325_finite__subset__induct_H,axiom,
    ! [F2: set_Extended_enat,A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( ord_le7203529160286727270d_enat @ F2 @ A2 )
       => ( ( P @ bot_bo7653980558646680370d_enat )
         => ( ! [A5: extended_enat,F3: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F3 )
               => ( ( member_Extended_enat @ A5 @ A2 )
                 => ( ( ord_le7203529160286727270d_enat @ F3 @ A2 )
                   => ( ~ ( member_Extended_enat @ A5 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_Extended_enat @ A5 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3326_finite__subset__induct_H,axiom,
    ! [F2: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( ord_less_eq_set_real @ F2 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A5: real,F3: set_real] :
                ( ( finite_finite_real @ F3 )
               => ( ( member_real @ A5 @ A2 )
                 => ( ( ord_less_eq_set_real @ F3 @ A2 )
                   => ( ~ ( member_real @ A5 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_real @ A5 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3327_finite__subset__induct_H,axiom,
    ! [F2: set_o,A2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( ord_less_eq_set_o @ F2 @ A2 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A5: $o,F3: set_o] :
                ( ( finite_finite_o @ F3 )
               => ( ( member_o @ A5 @ A2 )
                 => ( ( ord_less_eq_set_o @ F3 @ A2 )
                   => ( ~ ( member_o @ A5 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_o @ A5 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3328_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A5 @ A2 )
                 => ( ( ord_less_eq_set_nat @ F3 @ A2 )
                   => ( ~ ( member_nat @ A5 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3329_finite__subset__induct_H,axiom,
    ! [F2: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( ord_less_eq_set_int @ F2 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A5: int,F3: set_int] :
                ( ( finite_finite_int @ F3 )
               => ( ( member_int @ A5 @ A2 )
                 => ( ( ord_less_eq_set_int @ F3 @ A2 )
                   => ( ~ ( member_int @ A5 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_int @ A5 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3330_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_3331_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_3332_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_3333_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_3334_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_3335_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_3336_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_3337_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_3338_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_3339_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_3340_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_3341_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_3342_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3343_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3344_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3345_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3346_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_3347_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_3348_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_3349_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_3350_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3351_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3352_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3353_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3354_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_3355_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_3356_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_3357_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_3358_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_3359_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_3360_card__insert__if,axiom,
    ! [A2: set_real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( finite_card_real @ ( insert_real @ X @ A2 ) )
            = ( finite_card_real @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( finite_card_real @ ( insert_real @ X @ A2 ) )
            = ( suc @ ( finite_card_real @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_3361_card__insert__if,axiom,
    ! [A2: set_o,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( finite_card_o @ ( insert_o @ X @ A2 ) )
            = ( finite_card_o @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( finite_card_o @ ( insert_o @ X @ A2 ) )
            = ( suc @ ( finite_card_o @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_3362_card__insert__if,axiom,
    ! [A2: set_set_nat_rat,X: set_nat_rat] :
      ( ( finite6430367030675640852at_rat @ A2 )
     => ( ( ( member_set_nat_rat @ X @ A2 )
         => ( ( finite8736671560171388117at_rat @ ( insert_set_nat_rat @ X @ A2 ) )
            = ( finite8736671560171388117at_rat @ A2 ) ) )
        & ( ~ ( member_set_nat_rat @ X @ A2 )
         => ( ( finite8736671560171388117at_rat @ ( insert_set_nat_rat @ X @ A2 ) )
            = ( suc @ ( finite8736671560171388117at_rat @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_3363_card__insert__if,axiom,
    ! [A2: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( ( member_list_nat @ X @ A2 )
         => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A2 ) )
            = ( finite_card_list_nat @ A2 ) ) )
        & ( ~ ( member_list_nat @ X @ A2 )
         => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A2 ) )
            = ( suc @ ( finite_card_list_nat @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_3364_card__insert__if,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( ( member_set_nat @ X @ A2 )
         => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A2 ) )
            = ( finite_card_set_nat @ A2 ) ) )
        & ( ~ ( member_set_nat @ X @ A2 )
         => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A2 ) )
            = ( suc @ ( finite_card_set_nat @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_3365_card__insert__if,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
            = ( finite_card_nat @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
            = ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_3366_card__insert__if,axiom,
    ! [A2: set_int,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( finite_card_int @ ( insert_int @ X @ A2 ) )
            = ( finite_card_int @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( finite_card_int @ ( insert_int @ X @ A2 ) )
            = ( suc @ ( finite_card_int @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_3367_card__insert__if,axiom,
    ! [A2: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( finite_card_complex @ ( insert_complex @ X @ A2 ) )
            = ( finite_card_complex @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( finite_card_complex @ ( insert_complex @ X @ A2 ) )
            = ( suc @ ( finite_card_complex @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_3368_card__insert__if,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ( member8440522571783428010at_nat @ X @ A2 )
         => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) )
            = ( finite711546835091564841at_nat @ A2 ) ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
         => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) )
            = ( suc @ ( finite711546835091564841at_nat @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_3369_card__insert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ X @ A2 )
         => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X @ A2 ) )
            = ( finite121521170596916366d_enat @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ X @ A2 )
         => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X @ A2 ) )
            = ( suc @ ( finite121521170596916366d_enat @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_3370_card__Suc__eq__finite,axiom,
    ! [A2: set_real,K: nat] :
      ( ( ( finite_card_real @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: real,B6: set_real] :
            ( ( A2
              = ( insert_real @ B4 @ B6 ) )
            & ~ ( member_real @ B4 @ B6 )
            & ( ( finite_card_real @ B6 )
              = K )
            & ( finite_finite_real @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_3371_card__Suc__eq__finite,axiom,
    ! [A2: set_o,K: nat] :
      ( ( ( finite_card_o @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: $o,B6: set_o] :
            ( ( A2
              = ( insert_o @ B4 @ B6 ) )
            & ~ ( member_o @ B4 @ B6 )
            & ( ( finite_card_o @ B6 )
              = K )
            & ( finite_finite_o @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_3372_card__Suc__eq__finite,axiom,
    ! [A2: set_set_nat_rat,K: nat] :
      ( ( ( finite8736671560171388117at_rat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: set_nat_rat,B6: set_set_nat_rat] :
            ( ( A2
              = ( insert_set_nat_rat @ B4 @ B6 ) )
            & ~ ( member_set_nat_rat @ B4 @ B6 )
            & ( ( finite8736671560171388117at_rat @ B6 )
              = K )
            & ( finite6430367030675640852at_rat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_3373_card__Suc__eq__finite,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( ( finite_card_list_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: list_nat,B6: set_list_nat] :
            ( ( A2
              = ( insert_list_nat @ B4 @ B6 ) )
            & ~ ( member_list_nat @ B4 @ B6 )
            & ( ( finite_card_list_nat @ B6 )
              = K )
            & ( finite8100373058378681591st_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_3374_card__Suc__eq__finite,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( ( finite_card_set_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: set_nat,B6: set_set_nat] :
            ( ( A2
              = ( insert_set_nat @ B4 @ B6 ) )
            & ~ ( member_set_nat @ B4 @ B6 )
            & ( ( finite_card_set_nat @ B6 )
              = K )
            & ( finite1152437895449049373et_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_3375_card__Suc__eq__finite,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: nat,B6: set_nat] :
            ( ( A2
              = ( insert_nat @ B4 @ B6 ) )
            & ~ ( member_nat @ B4 @ B6 )
            & ( ( finite_card_nat @ B6 )
              = K )
            & ( finite_finite_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_3376_card__Suc__eq__finite,axiom,
    ! [A2: set_int,K: nat] :
      ( ( ( finite_card_int @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: int,B6: set_int] :
            ( ( A2
              = ( insert_int @ B4 @ B6 ) )
            & ~ ( member_int @ B4 @ B6 )
            & ( ( finite_card_int @ B6 )
              = K )
            & ( finite_finite_int @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_3377_card__Suc__eq__finite,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( ( finite_card_complex @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: complex,B6: set_complex] :
            ( ( A2
              = ( insert_complex @ B4 @ B6 ) )
            & ~ ( member_complex @ B4 @ B6 )
            & ( ( finite_card_complex @ B6 )
              = K )
            & ( finite3207457112153483333omplex @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_3378_card__Suc__eq__finite,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( ( finite711546835091564841at_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
            ( ( A2
              = ( insert8211810215607154385at_nat @ B4 @ B6 ) )
            & ~ ( member8440522571783428010at_nat @ B4 @ B6 )
            & ( ( finite711546835091564841at_nat @ B6 )
              = K )
            & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_3379_card__Suc__eq__finite,axiom,
    ! [A2: set_Extended_enat,K: nat] :
      ( ( ( finite121521170596916366d_enat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: extended_enat,B6: set_Extended_enat] :
            ( ( A2
              = ( insert_Extended_enat @ B4 @ B6 ) )
            & ~ ( member_Extended_enat @ B4 @ B6 )
            & ( ( finite121521170596916366d_enat @ B6 )
              = K )
            & ( finite4001608067531595151d_enat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_3380_infinite__remove,axiom,
    ! [S2: set_complex,A: complex] :
      ( ~ ( finite3207457112153483333omplex @ S2 )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_3381_infinite__remove,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S2 )
     => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% infinite_remove
thf(fact_3382_infinite__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ S2 )
     => ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% infinite_remove
thf(fact_3383_infinite__remove,axiom,
    ! [S2: set_real,A: real] :
      ( ~ ( finite_finite_real @ S2 )
     => ~ ( finite_finite_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% infinite_remove
thf(fact_3384_infinite__remove,axiom,
    ! [S2: set_o,A: $o] :
      ( ~ ( finite_finite_o @ S2 )
     => ~ ( finite_finite_o @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% infinite_remove
thf(fact_3385_infinite__remove,axiom,
    ! [S2: set_int,A: int] :
      ( ~ ( finite_finite_int @ S2 )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_3386_infinite__remove,axiom,
    ! [S2: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_3387_infinite__coinduct,axiom,
    ! [X5: set_complex > $o,A2: set_complex] :
      ( ( X5 @ A2 )
     => ( ! [A3: set_complex] :
            ( ( X5 @ A3 )
           => ? [X2: complex] :
                ( ( member_complex @ X2 @ A3 )
                & ( ( X5 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_3388_infinite__coinduct,axiom,
    ! [X5: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
      ( ( X5 @ A2 )
     => ( ! [A3: set_Pr1261947904930325089at_nat] :
            ( ( X5 @ A3 )
           => ? [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ A3 )
                & ( ( X5 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
                  | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
       => ~ ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_3389_infinite__coinduct,axiom,
    ! [X5: set_Extended_enat > $o,A2: set_Extended_enat] :
      ( ( X5 @ A2 )
     => ( ! [A3: set_Extended_enat] :
            ( ( X5 @ A3 )
           => ? [X2: extended_enat] :
                ( ( member_Extended_enat @ X2 @ A3 )
                & ( ( X5 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) )
                  | ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
       => ~ ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_3390_infinite__coinduct,axiom,
    ! [X5: set_real > $o,A2: set_real] :
      ( ( X5 @ A2 )
     => ( ! [A3: set_real] :
            ( ( X5 @ A3 )
           => ? [X2: real] :
                ( ( member_real @ X2 @ A3 )
                & ( ( X5 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_3391_infinite__coinduct,axiom,
    ! [X5: set_o > $o,A2: set_o] :
      ( ( X5 @ A2 )
     => ( ! [A3: set_o] :
            ( ( X5 @ A3 )
           => ? [X2: $o] :
                ( ( member_o @ X2 @ A3 )
                & ( ( X5 @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
                  | ~ ( finite_finite_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) )
       => ~ ( finite_finite_o @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_3392_infinite__coinduct,axiom,
    ! [X5: set_int > $o,A2: set_int] :
      ( ( X5 @ A2 )
     => ( ! [A3: set_int] :
            ( ( X5 @ A3 )
           => ? [X2: int] :
                ( ( member_int @ X2 @ A3 )
                & ( ( X5 @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_3393_infinite__coinduct,axiom,
    ! [X5: set_nat > $o,A2: set_nat] :
      ( ( X5 @ A2 )
     => ( ! [A3: set_nat] :
            ( ( X5 @ A3 )
           => ? [X2: nat] :
                ( ( member_nat @ X2 @ A3 )
                & ( ( X5 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_3394_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: set_nat,A3: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A3 )
             => ( ( member_set_nat @ A5 @ A3 )
               => ( ( P @ A3 )
                 => ( P @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A5 @ bot_bot_set_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_3395_finite__empty__induct,axiom,
    ! [A2: set_set_nat_rat,P: set_set_nat_rat > $o] :
      ( ( finite6430367030675640852at_rat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: set_nat_rat,A3: set_set_nat_rat] :
              ( ( finite6430367030675640852at_rat @ A3 )
             => ( ( member_set_nat_rat @ A5 @ A3 )
               => ( ( P @ A3 )
                 => ( P @ ( minus_1626877696091177228at_rat @ A3 @ ( insert_set_nat_rat @ A5 @ bot_bo6797373522285170759at_rat ) ) ) ) ) )
         => ( P @ bot_bo6797373522285170759at_rat ) ) ) ) ).

% finite_empty_induct
thf(fact_3396_finite__empty__induct,axiom,
    ! [A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: complex,A3: set_complex] :
              ( ( finite3207457112153483333omplex @ A3 )
             => ( ( member_complex @ A5 @ A3 )
               => ( ( P @ A3 )
                 => ( P @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A5 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_3397_finite__empty__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A3 )
             => ( ( member8440522571783428010at_nat @ A5 @ A3 )
               => ( ( P @ A3 )
                 => ( P @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A5 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
         => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_3398_finite__empty__induct,axiom,
    ! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: extended_enat,A3: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A3 )
             => ( ( member_Extended_enat @ A5 @ A3 )
               => ( ( P @ A3 )
                 => ( P @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A5 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
         => ( P @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% finite_empty_induct
thf(fact_3399_finite__empty__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: real,A3: set_real] :
              ( ( finite_finite_real @ A3 )
             => ( ( member_real @ A5 @ A3 )
               => ( ( P @ A3 )
                 => ( P @ ( minus_minus_set_real @ A3 @ ( insert_real @ A5 @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_3400_finite__empty__induct,axiom,
    ! [A2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: $o,A3: set_o] :
              ( ( finite_finite_o @ A3 )
             => ( ( member_o @ A5 @ A3 )
               => ( ( P @ A3 )
                 => ( P @ ( minus_minus_set_o @ A3 @ ( insert_o @ A5 @ bot_bot_set_o ) ) ) ) ) )
         => ( P @ bot_bot_set_o ) ) ) ) ).

% finite_empty_induct
thf(fact_3401_finite__empty__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: int,A3: set_int] :
              ( ( finite_finite_int @ A3 )
             => ( ( member_int @ A5 @ A3 )
               => ( ( P @ A3 )
                 => ( P @ ( minus_minus_set_int @ A3 @ ( insert_int @ A5 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_3402_finite__empty__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: nat,A3: set_nat] :
              ( ( finite_finite_nat @ A3 )
             => ( ( member_nat @ A5 @ A3 )
               => ( ( P @ A3 )
                 => ( P @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_3403_Diff__single__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 )
     => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_3404_Diff__single__insert,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_3405_Diff__single__insert,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 )
     => ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_3406_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_3407_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_3408_subset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
      = ( ( ( member8440522571783428010at_nat @ X @ A2 )
         => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
         => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_3409_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
      = ( ( ( member_set_nat @ X @ A2 )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
        & ( ~ ( member_set_nat @ X @ A2 )
         => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_3410_subset__insert__iff,axiom,
    ! [A2: set_set_nat_rat,X: set_nat_rat,B2: set_set_nat_rat] :
      ( ( ord_le4375437777232675859at_rat @ A2 @ ( insert_set_nat_rat @ X @ B2 ) )
      = ( ( ( member_set_nat_rat @ X @ A2 )
         => ( ord_le4375437777232675859at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) ) @ B2 ) )
        & ( ~ ( member_set_nat_rat @ X @ A2 )
         => ( ord_le4375437777232675859at_rat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_3411_subset__insert__iff,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
      = ( ( ( member_real @ X @ A2 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_3412_subset__insert__iff,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
      = ( ( ( member_o @ X @ A2 )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_3413_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
      = ( ( ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_3414_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
      = ( ( ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_3415_card__1__singletonE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A2 )
        = one_one_nat )
     => ~ ! [X4: product_prod_nat_nat] :
            ( A2
           != ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ).

% card_1_singletonE
thf(fact_3416_card__1__singletonE,axiom,
    ! [A2: set_complex] :
      ( ( ( finite_card_complex @ A2 )
        = one_one_nat )
     => ~ ! [X4: complex] :
            ( A2
           != ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ).

% card_1_singletonE
thf(fact_3417_card__1__singletonE,axiom,
    ! [A2: set_list_nat] :
      ( ( ( finite_card_list_nat @ A2 )
        = one_one_nat )
     => ~ ! [X4: list_nat] :
            ( A2
           != ( insert_list_nat @ X4 @ bot_bot_set_list_nat ) ) ) ).

% card_1_singletonE
thf(fact_3418_card__1__singletonE,axiom,
    ! [A2: set_set_nat] :
      ( ( ( finite_card_set_nat @ A2 )
        = one_one_nat )
     => ~ ! [X4: set_nat] :
            ( A2
           != ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_3419_card__1__singletonE,axiom,
    ! [A2: set_real] :
      ( ( ( finite_card_real @ A2 )
        = one_one_nat )
     => ~ ! [X4: real] :
            ( A2
           != ( insert_real @ X4 @ bot_bot_set_real ) ) ) ).

% card_1_singletonE
thf(fact_3420_card__1__singletonE,axiom,
    ! [A2: set_o] :
      ( ( ( finite_card_o @ A2 )
        = one_one_nat )
     => ~ ! [X4: $o] :
            ( A2
           != ( insert_o @ X4 @ bot_bot_set_o ) ) ) ).

% card_1_singletonE
thf(fact_3421_card__1__singletonE,axiom,
    ! [A2: set_nat] :
      ( ( ( finite_card_nat @ A2 )
        = one_one_nat )
     => ~ ! [X4: nat] :
            ( A2
           != ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_3422_card__1__singletonE,axiom,
    ! [A2: set_int] :
      ( ( ( finite_card_int @ A2 )
        = one_one_nat )
     => ~ ! [X4: int] :
            ( A2
           != ( insert_int @ X4 @ bot_bot_set_int ) ) ) ).

% card_1_singletonE
thf(fact_3423_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% one_less_mult
thf(fact_3424_n__less__m__mult__n,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_3425_n__less__n__mult__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).

% n_less_n_mult_m
thf(fact_3426_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q4 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q4 ) @ N )
        = ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q4 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_3427_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_3428_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
      = ( ord_less_int @ W2 @ Z ) ) ).

% add1_zle_eq
thf(fact_3429_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I2: int] :
            ( ( ord_less_eq_int @ K @ I2 )
           => ( ( P @ I2 )
             => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_3430_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W2 @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_3431_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W2 )
        | ( ord_less_eq_int @ zero_zero_int @ Z ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_eq_int @ W2 @ Z ) ) ) ).

% nat_le_eq_zle
thf(fact_3432_nat__eq__iff,axiom,
    ! [W2: int,M2: nat] :
      ( ( ( nat2 @ W2 )
        = M2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M2 ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_3433_nat__eq__iff2,axiom,
    ! [M2: nat,W2: int] :
      ( ( M2
        = ( nat2 @ W2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M2 ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_3434_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N4: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ( P @ N4 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_3435_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_3436_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
          = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_3437_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
     => ( ( ord_less_eq_int @ Z6 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_3438_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_3439_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_3440_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z3: real] :
          ( ( ord_less_real @ zero_zero_real @ Z3 )
         => ( ( ord_less_real @ Z3 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3441_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: 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 ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3442_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3443_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_3444_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_3445_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3446_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_3447_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_3448_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3449_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_3450_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_3451_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3452_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_3453_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_3454_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3455_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_3456_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_3457_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3458_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_3459_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_3460_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3461_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_3462_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_3463_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3464_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_3465_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_3466_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3467_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_3468_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3469_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_3470_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3471_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_3472_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3473_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_3474_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3475_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_3476_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3477_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_3478_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3479_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_3480_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3481_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3482_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3483_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3484_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3485_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_3486_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_3487_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_3488_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_3489_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_3490_card__Suc__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( ( finite711546835091564841at_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
            ( ( A2
              = ( insert8211810215607154385at_nat @ B4 @ B6 ) )
            & ~ ( member8440522571783428010at_nat @ B4 @ B6 )
            & ( ( finite711546835091564841at_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_3491_card__Suc__eq,axiom,
    ! [A2: set_set_nat_rat,K: nat] :
      ( ( ( finite8736671560171388117at_rat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: set_nat_rat,B6: set_set_nat_rat] :
            ( ( A2
              = ( insert_set_nat_rat @ B4 @ B6 ) )
            & ~ ( member_set_nat_rat @ B4 @ B6 )
            & ( ( finite8736671560171388117at_rat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bo6797373522285170759at_rat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_3492_card__Suc__eq,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( ( finite_card_complex @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: complex,B6: set_complex] :
            ( ( A2
              = ( insert_complex @ B4 @ B6 ) )
            & ~ ( member_complex @ B4 @ B6 )
            & ( ( finite_card_complex @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_complex ) ) ) ) ) ).

% card_Suc_eq
thf(fact_3493_card__Suc__eq,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( ( finite_card_list_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: list_nat,B6: set_list_nat] :
            ( ( A2
              = ( insert_list_nat @ B4 @ B6 ) )
            & ~ ( member_list_nat @ B4 @ B6 )
            & ( ( finite_card_list_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_list_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_3494_card__Suc__eq,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( ( finite_card_set_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: set_nat,B6: set_set_nat] :
            ( ( A2
              = ( insert_set_nat @ B4 @ B6 ) )
            & ~ ( member_set_nat @ B4 @ B6 )
            & ( ( finite_card_set_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_set_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_3495_card__Suc__eq,axiom,
    ! [A2: set_real,K: nat] :
      ( ( ( finite_card_real @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: real,B6: set_real] :
            ( ( A2
              = ( insert_real @ B4 @ B6 ) )
            & ~ ( member_real @ B4 @ B6 )
            & ( ( finite_card_real @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_real ) ) ) ) ) ).

% card_Suc_eq
thf(fact_3496_card__Suc__eq,axiom,
    ! [A2: set_o,K: nat] :
      ( ( ( finite_card_o @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: $o,B6: set_o] :
            ( ( A2
              = ( insert_o @ B4 @ B6 ) )
            & ~ ( member_o @ B4 @ B6 )
            & ( ( finite_card_o @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_o ) ) ) ) ) ).

% card_Suc_eq
thf(fact_3497_card__Suc__eq,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: nat,B6: set_nat] :
            ( ( A2
              = ( insert_nat @ B4 @ B6 ) )
            & ~ ( member_nat @ B4 @ B6 )
            & ( ( finite_card_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_3498_card__Suc__eq,axiom,
    ! [A2: set_int,K: nat] :
      ( ( ( finite_card_int @ A2 )
        = ( suc @ K ) )
      = ( ? [B4: int,B6: set_int] :
            ( ( A2
              = ( insert_int @ B4 @ B6 ) )
            & ~ ( member_int @ B4 @ B6 )
            & ( ( finite_card_int @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_int ) ) ) ) ) ).

% card_Suc_eq
thf(fact_3499_card__eq__SucD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( ( finite711546835091564841at_nat @ A2 )
        = ( suc @ K ) )
     => ? [B5: product_prod_nat_nat,B8: set_Pr1261947904930325089at_nat] :
          ( ( A2
            = ( insert8211810215607154385at_nat @ B5 @ B8 ) )
          & ~ ( member8440522571783428010at_nat @ B5 @ B8 )
          & ( ( finite711546835091564841at_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bo2099793752762293965at_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_3500_card__eq__SucD,axiom,
    ! [A2: set_set_nat_rat,K: nat] :
      ( ( ( finite8736671560171388117at_rat @ A2 )
        = ( suc @ K ) )
     => ? [B5: set_nat_rat,B8: set_set_nat_rat] :
          ( ( A2
            = ( insert_set_nat_rat @ B5 @ B8 ) )
          & ~ ( member_set_nat_rat @ B5 @ B8 )
          & ( ( finite8736671560171388117at_rat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bo6797373522285170759at_rat ) ) ) ) ).

% card_eq_SucD
thf(fact_3501_card__eq__SucD,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( ( finite_card_complex @ A2 )
        = ( suc @ K ) )
     => ? [B5: complex,B8: set_complex] :
          ( ( A2
            = ( insert_complex @ B5 @ B8 ) )
          & ~ ( member_complex @ B5 @ B8 )
          & ( ( finite_card_complex @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_complex ) ) ) ) ).

% card_eq_SucD
thf(fact_3502_card__eq__SucD,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( ( finite_card_list_nat @ A2 )
        = ( suc @ K ) )
     => ? [B5: list_nat,B8: set_list_nat] :
          ( ( A2
            = ( insert_list_nat @ B5 @ B8 ) )
          & ~ ( member_list_nat @ B5 @ B8 )
          & ( ( finite_card_list_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_list_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_3503_card__eq__SucD,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( ( finite_card_set_nat @ A2 )
        = ( suc @ K ) )
     => ? [B5: set_nat,B8: set_set_nat] :
          ( ( A2
            = ( insert_set_nat @ B5 @ B8 ) )
          & ~ ( member_set_nat @ B5 @ B8 )
          & ( ( finite_card_set_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_set_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_3504_card__eq__SucD,axiom,
    ! [A2: set_real,K: nat] :
      ( ( ( finite_card_real @ A2 )
        = ( suc @ K ) )
     => ? [B5: real,B8: set_real] :
          ( ( A2
            = ( insert_real @ B5 @ B8 ) )
          & ~ ( member_real @ B5 @ B8 )
          & ( ( finite_card_real @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_real ) ) ) ) ).

% card_eq_SucD
thf(fact_3505_card__eq__SucD,axiom,
    ! [A2: set_o,K: nat] :
      ( ( ( finite_card_o @ A2 )
        = ( suc @ K ) )
     => ? [B5: $o,B8: set_o] :
          ( ( A2
            = ( insert_o @ B5 @ B8 ) )
          & ~ ( member_o @ B5 @ B8 )
          & ( ( finite_card_o @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_o ) ) ) ) ).

% card_eq_SucD
thf(fact_3506_card__eq__SucD,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A2 )
        = ( suc @ K ) )
     => ? [B5: nat,B8: set_nat] :
          ( ( A2
            = ( insert_nat @ B5 @ B8 ) )
          & ~ ( member_nat @ B5 @ B8 )
          & ( ( finite_card_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_3507_card__eq__SucD,axiom,
    ! [A2: set_int,K: nat] :
      ( ( ( finite_card_int @ A2 )
        = ( suc @ K ) )
     => ? [B5: int,B8: set_int] :
          ( ( A2
            = ( insert_int @ B5 @ B8 ) )
          & ~ ( member_int @ B5 @ B8 )
          & ( ( finite_card_int @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
thf(fact_3508_card__1__singleton__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A2 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X3: product_prod_nat_nat] :
            ( A2
            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_3509_card__1__singleton__iff,axiom,
    ! [A2: set_complex] :
      ( ( ( finite_card_complex @ A2 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X3: complex] :
            ( A2
            = ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ).

% card_1_singleton_iff
thf(fact_3510_card__1__singleton__iff,axiom,
    ! [A2: set_list_nat] :
      ( ( ( finite_card_list_nat @ A2 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X3: list_nat] :
            ( A2
            = ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_3511_card__1__singleton__iff,axiom,
    ! [A2: set_set_nat] :
      ( ( ( finite_card_set_nat @ A2 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X3: set_nat] :
            ( A2
            = ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_3512_card__1__singleton__iff,axiom,
    ! [A2: set_real] :
      ( ( ( finite_card_real @ A2 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X3: real] :
            ( A2
            = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).

% card_1_singleton_iff
thf(fact_3513_card__1__singleton__iff,axiom,
    ! [A2: set_o] :
      ( ( ( finite_card_o @ A2 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X3: $o] :
            ( A2
            = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).

% card_1_singleton_iff
thf(fact_3514_card__1__singleton__iff,axiom,
    ! [A2: set_nat] :
      ( ( ( finite_card_nat @ A2 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X3: nat] :
            ( A2
            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_3515_card__1__singleton__iff,axiom,
    ! [A2: set_int] :
      ( ( ( finite_card_int @ A2 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X3: int] :
            ( A2
            = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).

% card_1_singleton_iff
thf(fact_3516_remove__induct,axiom,
    ! [P: set_set_nat > $o,B2: set_set_nat] :
      ( ( P @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A3: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A3 )
             => ( ( A3 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
                 => ( ! [X2: set_nat] :
                        ( ( member_set_nat @ X2 @ A3 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_3517_remove__induct,axiom,
    ! [P: set_set_nat_rat > $o,B2: set_set_nat_rat] :
      ( ( P @ bot_bo6797373522285170759at_rat )
     => ( ( ~ ( finite6430367030675640852at_rat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A3: set_set_nat_rat] :
              ( ( finite6430367030675640852at_rat @ A3 )
             => ( ( A3 != bot_bo6797373522285170759at_rat )
               => ( ( ord_le4375437777232675859at_rat @ A3 @ B2 )
                 => ( ! [X2: set_nat_rat] :
                        ( ( member_set_nat_rat @ X2 @ A3 )
                       => ( P @ ( minus_1626877696091177228at_rat @ A3 @ ( insert_set_nat_rat @ X2 @ bot_bo6797373522285170759at_rat ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_3518_remove__induct,axiom,
    ! [P: set_complex > $o,B2: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B2 )
         => ( P @ B2 ) )
       => ( ! [A3: set_complex] :
              ( ( finite3207457112153483333omplex @ A3 )
             => ( ( A3 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
                 => ( ! [X2: complex] :
                        ( ( member_complex @ X2 @ A3 )
                       => ( P @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_3519_remove__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,B2: set_Pr1261947904930325089at_nat] :
      ( ( P @ bot_bo2099793752762293965at_nat )
     => ( ( ~ ( finite6177210948735845034at_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A3: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A3 )
             => ( ( A3 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
                 => ( ! [X2: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X2 @ A3 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_3520_remove__induct,axiom,
    ! [P: set_Extended_enat > $o,B2: set_Extended_enat] :
      ( ( P @ bot_bo7653980558646680370d_enat )
     => ( ( ~ ( finite4001608067531595151d_enat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A3: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A3 )
             => ( ( A3 != bot_bo7653980558646680370d_enat )
               => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
                 => ( ! [X2: extended_enat] :
                        ( ( member_Extended_enat @ X2 @ A3 )
                       => ( P @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_3521_remove__induct,axiom,
    ! [P: set_real > $o,B2: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B2 )
         => ( P @ B2 ) )
       => ( ! [A3: set_real] :
              ( ( finite_finite_real @ A3 )
             => ( ( A3 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A3 @ B2 )
                 => ( ! [X2: real] :
                        ( ( member_real @ X2 @ A3 )
                       => ( P @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_3522_remove__induct,axiom,
    ! [P: set_o > $o,B2: set_o] :
      ( ( P @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B2 )
         => ( P @ B2 ) )
       => ( ! [A3: set_o] :
              ( ( finite_finite_o @ A3 )
             => ( ( A3 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A3 @ B2 )
                 => ( ! [X2: $o] :
                        ( ( member_o @ X2 @ A3 )
                       => ( P @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_3523_remove__induct,axiom,
    ! [P: set_nat > $o,B2: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A3: set_nat] :
              ( ( finite_finite_nat @ A3 )
             => ( ( A3 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A3 @ B2 )
                 => ( ! [X2: nat] :
                        ( ( member_nat @ X2 @ A3 )
                       => ( P @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_3524_remove__induct,axiom,
    ! [P: set_int > $o,B2: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B2 )
         => ( P @ B2 ) )
       => ( ! [A3: set_int] :
              ( ( finite_finite_int @ A3 )
             => ( ( A3 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A3 @ B2 )
                 => ( ! [X2: int] :
                        ( ( member_int @ X2 @ A3 )
                       => ( P @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_3525_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [A3: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A3 )
             => ( ( A3 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
                 => ( ! [X2: set_nat] :
                        ( ( member_set_nat @ X2 @ A3 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_3526_finite__remove__induct,axiom,
    ! [B2: set_set_nat_rat,P: set_set_nat_rat > $o] :
      ( ( finite6430367030675640852at_rat @ B2 )
     => ( ( P @ bot_bo6797373522285170759at_rat )
       => ( ! [A3: set_set_nat_rat] :
              ( ( finite6430367030675640852at_rat @ A3 )
             => ( ( A3 != bot_bo6797373522285170759at_rat )
               => ( ( ord_le4375437777232675859at_rat @ A3 @ B2 )
                 => ( ! [X2: set_nat_rat] :
                        ( ( member_set_nat_rat @ X2 @ A3 )
                       => ( P @ ( minus_1626877696091177228at_rat @ A3 @ ( insert_set_nat_rat @ X2 @ bot_bo6797373522285170759at_rat ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_3527_finite__remove__induct,axiom,
    ! [B2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A3: set_complex] :
              ( ( finite3207457112153483333omplex @ A3 )
             => ( ( A3 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
                 => ( ! [X2: complex] :
                        ( ( member_complex @ X2 @ A3 )
                       => ( P @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_3528_finite__remove__induct,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [A3: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A3 )
             => ( ( A3 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
                 => ( ! [X2: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X2 @ A3 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_3529_finite__remove__induct,axiom,
    ! [B2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [A3: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A3 )
             => ( ( A3 != bot_bo7653980558646680370d_enat )
               => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
                 => ( ! [X2: extended_enat] :
                        ( ( member_Extended_enat @ X2 @ A3 )
                       => ( P @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_3530_finite__remove__induct,axiom,
    ! [B2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A3: set_real] :
              ( ( finite_finite_real @ A3 )
             => ( ( A3 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A3 @ B2 )
                 => ( ! [X2: real] :
                        ( ( member_real @ X2 @ A3 )
                       => ( P @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_3531_finite__remove__induct,axiom,
    ! [B2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [A3: set_o] :
              ( ( finite_finite_o @ A3 )
             => ( ( A3 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A3 @ B2 )
                 => ( ! [X2: $o] :
                        ( ( member_o @ X2 @ A3 )
                       => ( P @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_3532_finite__remove__induct,axiom,
    ! [B2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A3: set_nat] :
              ( ( finite_finite_nat @ A3 )
             => ( ( A3 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A3 @ B2 )
                 => ( ! [X2: nat] :
                        ( ( member_nat @ X2 @ A3 )
                       => ( P @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_3533_finite__remove__induct,axiom,
    ! [B2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A3: set_int] :
              ( ( finite_finite_int @ A3 )
             => ( ( A3 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A3 @ B2 )
                 => ( ! [X2: int] :
                        ( ( member_int @ X2 @ A3 )
                       => ( P @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
                   => ( P @ A3 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_3534_card__le__Suc__iff,axiom,
    ! [N: nat,A2: set_real] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_real @ A2 ) )
      = ( ? [A4: real,B6: set_real] :
            ( ( A2
              = ( insert_real @ A4 @ B6 ) )
            & ~ ( member_real @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_real @ B6 ) )
            & ( finite_finite_real @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_3535_card__le__Suc__iff,axiom,
    ! [N: nat,A2: set_o] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_o @ A2 ) )
      = ( ? [A4: $o,B6: set_o] :
            ( ( A2
              = ( insert_o @ A4 @ B6 ) )
            & ~ ( member_o @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_o @ B6 ) )
            & ( finite_finite_o @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_3536_card__le__Suc__iff,axiom,
    ! [N: nat,A2: set_set_nat_rat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite8736671560171388117at_rat @ A2 ) )
      = ( ? [A4: set_nat_rat,B6: set_set_nat_rat] :
            ( ( A2
              = ( insert_set_nat_rat @ A4 @ B6 ) )
            & ~ ( member_set_nat_rat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite8736671560171388117at_rat @ B6 ) )
            & ( finite6430367030675640852at_rat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_3537_card__le__Suc__iff,axiom,
    ! [N: nat,A2: set_list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_list_nat @ A2 ) )
      = ( ? [A4: list_nat,B6: set_list_nat] :
            ( ( A2
              = ( insert_list_nat @ A4 @ B6 ) )
            & ~ ( member_list_nat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ B6 ) )
            & ( finite8100373058378681591st_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_3538_card__le__Suc__iff,axiom,
    ! [N: nat,A2: set_set_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_set_nat @ A2 ) )
      = ( ? [A4: set_nat,B6: set_set_nat] :
            ( ( A2
              = ( insert_set_nat @ A4 @ B6 ) )
            & ~ ( member_set_nat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ B6 ) )
            & ( finite1152437895449049373et_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_3539_card__le__Suc__iff,axiom,
    ! [N: nat,A2: set_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A2 ) )
      = ( ? [A4: nat,B6: set_nat] :
            ( ( A2
              = ( insert_nat @ A4 @ B6 ) )
            & ~ ( member_nat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_nat @ B6 ) )
            & ( finite_finite_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_3540_card__le__Suc__iff,axiom,
    ! [N: nat,A2: set_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_int @ A2 ) )
      = ( ? [A4: int,B6: set_int] :
            ( ( A2
              = ( insert_int @ A4 @ B6 ) )
            & ~ ( member_int @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_int @ B6 ) )
            & ( finite_finite_int @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_3541_card__le__Suc__iff,axiom,
    ! [N: nat,A2: set_complex] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_complex @ A2 ) )
      = ( ? [A4: complex,B6: set_complex] :
            ( ( A2
              = ( insert_complex @ A4 @ B6 ) )
            & ~ ( member_complex @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_complex @ B6 ) )
            & ( finite3207457112153483333omplex @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_3542_card__le__Suc__iff,axiom,
    ! [N: nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite711546835091564841at_nat @ A2 ) )
      = ( ? [A4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
            ( ( A2
              = ( insert8211810215607154385at_nat @ A4 @ B6 ) )
            & ~ ( member8440522571783428010at_nat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ B6 ) )
            & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_3543_card__le__Suc__iff,axiom,
    ! [N: nat,A2: set_Extended_enat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite121521170596916366d_enat @ A2 ) )
      = ( ? [A4: extended_enat,B6: set_Extended_enat] :
            ( ( A2
              = ( insert_Extended_enat @ A4 @ B6 ) )
            & ~ ( member_Extended_enat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ B6 ) )
            & ( finite4001608067531595151d_enat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_3544_card__Diff1__le,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A2 ) ) ).

% card_Diff1_le
thf(fact_3545_card__Diff1__le,axiom,
    ! [A2: set_complex,X: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A2 ) ) ).

% card_Diff1_le
thf(fact_3546_card__Diff1__le,axiom,
    ! [A2: set_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A2 ) ) ).

% card_Diff1_le
thf(fact_3547_card__Diff1__le,axiom,
    ! [A2: set_set_nat,X: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) ) ).

% card_Diff1_le
thf(fact_3548_card__Diff1__le,axiom,
    ! [A2: set_real,X: real] : ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ).

% card_Diff1_le
thf(fact_3549_card__Diff1__le,axiom,
    ! [A2: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A2 ) ) ).

% card_Diff1_le
thf(fact_3550_card__Diff1__le,axiom,
    ! [A2: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A2 ) ) ).

% card_Diff1_le
thf(fact_3551_card__Diff1__le,axiom,
    ! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).

% card_Diff1_le
thf(fact_3552_finite__induct__select,axiom,
    ! [S2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T4: set_complex] :
              ( ( ord_less_set_complex @ T4 @ S2 )
             => ( ( P @ T4 )
               => ? [X2: complex] :
                    ( ( member_complex @ X2 @ ( minus_811609699411566653omplex @ S2 @ T4 ) )
                    & ( P @ ( insert_complex @ X2 @ T4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_3553_finite__induct__select,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [T4: set_Pr1261947904930325089at_nat] :
              ( ( ord_le7866589430770878221at_nat @ T4 @ S2 )
             => ( ( P @ T4 )
               => ? [X2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X2 @ ( minus_1356011639430497352at_nat @ S2 @ T4 ) )
                    & ( P @ ( insert8211810215607154385at_nat @ X2 @ T4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_3554_finite__induct__select,axiom,
    ! [S2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [T4: set_Extended_enat] :
              ( ( ord_le2529575680413868914d_enat @ T4 @ S2 )
             => ( ( P @ T4 )
               => ? [X2: extended_enat] :
                    ( ( member_Extended_enat @ X2 @ ( minus_925952699566721837d_enat @ S2 @ T4 ) )
                    & ( P @ ( insert_Extended_enat @ X2 @ T4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_3555_finite__induct__select,axiom,
    ! [S2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ S2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [T4: set_real] :
              ( ( ord_less_set_real @ T4 @ S2 )
             => ( ( P @ T4 )
               => ? [X2: real] :
                    ( ( member_real @ X2 @ ( minus_minus_set_real @ S2 @ T4 ) )
                    & ( P @ ( insert_real @ X2 @ T4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_3556_finite__induct__select,axiom,
    ! [S2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ S2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [T4: set_o] :
              ( ( ord_less_set_o @ T4 @ S2 )
             => ( ( P @ T4 )
               => ? [X2: $o] :
                    ( ( member_o @ X2 @ ( minus_minus_set_o @ S2 @ T4 ) )
                    & ( P @ ( insert_o @ X2 @ T4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_3557_finite__induct__select,axiom,
    ! [S2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T4: set_int] :
              ( ( ord_less_set_int @ T4 @ S2 )
             => ( ( P @ T4 )
               => ? [X2: int] :
                    ( ( member_int @ X2 @ ( minus_minus_set_int @ S2 @ T4 ) )
                    & ( P @ ( insert_int @ X2 @ T4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_3558_finite__induct__select,axiom,
    ! [S2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T4: set_nat] :
              ( ( ord_less_set_nat @ T4 @ S2 )
             => ( ( P @ T4 )
               => ? [X2: nat] :
                    ( ( member_nat @ X2 @ ( minus_minus_set_nat @ S2 @ T4 ) )
                    & ( P @ ( insert_nat @ X2 @ T4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_3559_psubset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
      = ( ( ( member8440522571783428010at_nat @ X @ B2 )
         => ( ord_le7866589430770878221at_nat @ A2 @ B2 ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ B2 )
         => ( ( ( member8440522571783428010at_nat @ X @ A2 )
             => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
            & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
             => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3560_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
      = ( ( ( member_set_nat @ X @ B2 )
         => ( ord_less_set_set_nat @ A2 @ B2 ) )
        & ( ~ ( member_set_nat @ X @ B2 )
         => ( ( ( member_set_nat @ X @ A2 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
            & ( ~ ( member_set_nat @ X @ A2 )
             => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3561_psubset__insert__iff,axiom,
    ! [A2: set_set_nat_rat,X: set_nat_rat,B2: set_set_nat_rat] :
      ( ( ord_le1311537459589289991at_rat @ A2 @ ( insert_set_nat_rat @ X @ B2 ) )
      = ( ( ( member_set_nat_rat @ X @ B2 )
         => ( ord_le1311537459589289991at_rat @ A2 @ B2 ) )
        & ( ~ ( member_set_nat_rat @ X @ B2 )
         => ( ( ( member_set_nat_rat @ X @ A2 )
             => ( ord_le1311537459589289991at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) ) @ B2 ) )
            & ( ~ ( member_set_nat_rat @ X @ A2 )
             => ( ord_le4375437777232675859at_rat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3562_psubset__insert__iff,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B2 ) )
      = ( ( ( member_real @ X @ B2 )
         => ( ord_less_set_real @ A2 @ B2 ) )
        & ( ~ ( member_real @ X @ B2 )
         => ( ( ( member_real @ X @ A2 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
            & ( ~ ( member_real @ X @ A2 )
             => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3563_psubset__insert__iff,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( ( ord_less_set_o @ A2 @ ( insert_o @ X @ B2 ) )
      = ( ( ( member_o @ X @ B2 )
         => ( ord_less_set_o @ A2 @ B2 ) )
        & ( ~ ( member_o @ X @ B2 )
         => ( ( ( member_o @ X @ A2 )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
            & ( ~ ( member_o @ X @ A2 )
             => ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3564_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
      = ( ( ( member_nat @ X @ B2 )
         => ( ord_less_set_nat @ A2 @ B2 ) )
        & ( ~ ( member_nat @ X @ B2 )
         => ( ( ( member_nat @ X @ A2 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
            & ( ~ ( member_nat @ X @ A2 )
             => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3565_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B2 ) )
      = ( ( ( member_int @ X @ B2 )
         => ( ord_less_set_int @ A2 @ B2 ) )
        & ( ~ ( member_int @ X @ B2 )
         => ( ( ( member_int @ X @ A2 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
            & ( ~ ( member_int @ X @ A2 )
             => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3566_div__nat__eqI,axiom,
    ! [N: nat,Q4: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M2 )
     => ( ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
       => ( ( divide_divide_nat @ M2 @ N )
          = Q4 ) ) ) ).

% div_nat_eqI
thf(fact_3567_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q4 )
     => ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q4 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q4 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_3568_le__imp__0__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).

% le_imp_0_less
thf(fact_3569_frac__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).

% frac_add
thf(fact_3570_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_3571_nat__less__iff,axiom,
    ! [W2: int,M2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M2 )
        = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).

% nat_less_iff
thf(fact_3572_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3573_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3574_card_Oremove,axiom,
    ! [A2: set_set_nat_rat,X: set_nat_rat] :
      ( ( finite6430367030675640852at_rat @ A2 )
     => ( ( member_set_nat_rat @ X @ A2 )
       => ( ( finite8736671560171388117at_rat @ A2 )
          = ( suc @ ( finite8736671560171388117at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3575_card_Oremove,axiom,
    ! [A2: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( member_list_nat @ X @ A2 )
       => ( ( finite_card_list_nat @ A2 )
          = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3576_card_Oremove,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ X @ A2 )
       => ( ( finite_card_set_nat @ A2 )
          = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3577_card_Oremove,axiom,
    ! [A2: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( finite_card_complex @ A2 )
          = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3578_card_Oremove,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( finite711546835091564841at_nat @ A2 )
          = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3579_card_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( finite121521170596916366d_enat @ A2 )
          = ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3580_card_Oremove,axiom,
    ! [A2: set_real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( finite_card_real @ A2 )
          = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3581_card_Oremove,axiom,
    ! [A2: set_o,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ( finite_card_o @ A2 )
          = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3582_card_Oremove,axiom,
    ! [A2: set_int,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( finite_card_int @ A2 )
          = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3583_card_Oremove,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( finite_card_nat @ A2 )
          = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3584_card_Oinsert__remove,axiom,
    ! [A2: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A2 ) )
        = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3585_card_Oinsert__remove,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A2 ) )
        = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3586_card_Oinsert__remove,axiom,
    ! [A2: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_card_complex @ ( insert_complex @ X @ A2 ) )
        = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3587_card_Oinsert__remove,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) )
        = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3588_card_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X @ A2 ) )
        = ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3589_card_Oinsert__remove,axiom,
    ! [A2: set_real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_card_real @ ( insert_real @ X @ A2 ) )
        = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3590_card_Oinsert__remove,axiom,
    ! [A2: set_o,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( finite_card_o @ ( insert_o @ X @ A2 ) )
        = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3591_card_Oinsert__remove,axiom,
    ! [A2: set_int,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_card_int @ ( insert_int @ X @ A2 ) )
        = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3592_card_Oinsert__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_card_nat @ ( insert_nat @ X @ A2 ) )
        = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3593_card__Suc__Diff1,axiom,
    ! [A2: set_set_nat_rat,X: set_nat_rat] :
      ( ( finite6430367030675640852at_rat @ A2 )
     => ( ( member_set_nat_rat @ X @ A2 )
       => ( ( suc @ ( finite8736671560171388117at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) ) ) )
          = ( finite8736671560171388117at_rat @ A2 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3594_card__Suc__Diff1,axiom,
    ! [A2: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( member_list_nat @ X @ A2 )
       => ( ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) )
          = ( finite_card_list_nat @ A2 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3595_card__Suc__Diff1,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ X @ A2 )
       => ( ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) )
          = ( finite_card_set_nat @ A2 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3596_card__Suc__Diff1,axiom,
    ! [A2: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) )
          = ( finite_card_complex @ A2 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3597_card__Suc__Diff1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) )
          = ( finite711546835091564841at_nat @ A2 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3598_card__Suc__Diff1,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) )
          = ( finite121521170596916366d_enat @ A2 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3599_card__Suc__Diff1,axiom,
    ! [A2: set_real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) )
          = ( finite_card_real @ A2 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3600_card__Suc__Diff1,axiom,
    ! [A2: set_o,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) )
          = ( finite_card_o @ A2 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3601_card__Suc__Diff1,axiom,
    ! [A2: set_int,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
          = ( finite_card_int @ A2 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3602_card__Suc__Diff1,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
          = ( finite_card_nat @ A2 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3603_card__Diff1__less__iff,axiom,
    ! [A2: set_set_nat_rat,X: set_nat_rat] :
      ( ( ord_less_nat @ ( finite8736671560171388117at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) ) ) @ ( finite8736671560171388117at_rat @ A2 ) )
      = ( ( finite6430367030675640852at_rat @ A2 )
        & ( member_set_nat_rat @ X @ A2 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3604_card__Diff1__less__iff,axiom,
    ! [A2: set_list_nat,X: list_nat] :
      ( ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A2 ) )
      = ( ( finite8100373058378681591st_nat @ A2 )
        & ( member_list_nat @ X @ A2 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3605_card__Diff1__less__iff,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) )
      = ( ( finite1152437895449049373et_nat @ A2 )
        & ( member_set_nat @ X @ A2 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3606_card__Diff1__less__iff,axiom,
    ! [A2: set_complex,X: complex] :
      ( ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A2 ) )
      = ( ( finite3207457112153483333omplex @ A2 )
        & ( member_complex @ X @ A2 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3607_card__Diff1__less__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A2 ) )
      = ( ( finite6177210948735845034at_nat @ A2 )
        & ( member8440522571783428010at_nat @ X @ A2 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3608_card__Diff1__less__iff,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A2 ) )
      = ( ( finite4001608067531595151d_enat @ A2 )
        & ( member_Extended_enat @ X @ A2 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3609_card__Diff1__less__iff,axiom,
    ! [A2: set_real,X: real] :
      ( ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) )
      = ( ( finite_finite_real @ A2 )
        & ( member_real @ X @ A2 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3610_card__Diff1__less__iff,axiom,
    ! [A2: set_o,X: $o] :
      ( ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A2 ) )
      = ( ( finite_finite_o @ A2 )
        & ( member_o @ X @ A2 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3611_card__Diff1__less__iff,axiom,
    ! [A2: set_int,X: int] :
      ( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A2 ) )
      = ( ( finite_finite_int @ A2 )
        & ( member_int @ X @ A2 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3612_card__Diff1__less__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
      = ( ( finite_finite_nat @ A2 )
        & ( member_nat @ X @ A2 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3613_card__Diff2__less,axiom,
    ! [A2: set_set_nat_rat,X: set_nat_rat,Y: set_nat_rat] :
      ( ( finite6430367030675640852at_rat @ A2 )
     => ( ( member_set_nat_rat @ X @ A2 )
       => ( ( member_set_nat_rat @ Y @ A2 )
         => ( ord_less_nat @ ( finite8736671560171388117at_rat @ ( minus_1626877696091177228at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) ) @ ( insert_set_nat_rat @ Y @ bot_bo6797373522285170759at_rat ) ) ) @ ( finite8736671560171388117at_rat @ A2 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3614_card__Diff2__less,axiom,
    ! [A2: set_list_nat,X: list_nat,Y: list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( member_list_nat @ X @ A2 )
       => ( ( member_list_nat @ Y @ A2 )
         => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) @ ( insert_list_nat @ Y @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A2 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3615_card__Diff2__less,axiom,
    ! [A2: set_set_nat,X: set_nat,Y: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ X @ A2 )
       => ( ( member_set_nat @ Y @ A2 )
         => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3616_card__Diff2__less,axiom,
    ! [A2: set_complex,X: complex,Y: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( member_complex @ Y @ A2 )
         => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ ( insert_complex @ Y @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A2 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3617_card__Diff2__less,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( member8440522571783428010at_nat @ Y @ A2 )
         => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A2 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3618_card__Diff2__less,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( member_Extended_enat @ Y @ A2 )
         => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) @ ( insert_Extended_enat @ Y @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A2 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3619_card__Diff2__less,axiom,
    ! [A2: set_real,X: real,Y: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( member_real @ Y @ A2 )
         => ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ ( insert_real @ Y @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3620_card__Diff2__less,axiom,
    ! [A2: set_o,X: $o,Y: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ( member_o @ Y @ A2 )
         => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( insert_o @ Y @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A2 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3621_card__Diff2__less,axiom,
    ! [A2: set_int,X: int,Y: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( member_int @ Y @ A2 )
         => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( insert_int @ Y @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A2 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3622_card__Diff2__less,axiom,
    ! [A2: set_nat,X: nat,Y: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( member_nat @ Y @ A2 )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3623_card__Diff1__less,axiom,
    ! [A2: set_set_nat_rat,X: set_nat_rat] :
      ( ( finite6430367030675640852at_rat @ A2 )
     => ( ( member_set_nat_rat @ X @ A2 )
       => ( ord_less_nat @ ( finite8736671560171388117at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) ) ) @ ( finite8736671560171388117at_rat @ A2 ) ) ) ) ).

% card_Diff1_less
thf(fact_3624_card__Diff1__less,axiom,
    ! [A2: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( member_list_nat @ X @ A2 )
       => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A2 ) ) ) ) ).

% card_Diff1_less
thf(fact_3625_card__Diff1__less,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ X @ A2 )
       => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).

% card_Diff1_less
thf(fact_3626_card__Diff1__less,axiom,
    ! [A2: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A2 ) ) ) ) ).

% card_Diff1_less
thf(fact_3627_card__Diff1__less,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( member8440522571783428010at_nat @ X @ A2 )
       => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A2 ) ) ) ) ).

% card_Diff1_less
thf(fact_3628_card__Diff1__less,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A2 ) ) ) ) ).

% card_Diff1_less
thf(fact_3629_card__Diff1__less,axiom,
    ! [A2: set_real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ) ) ).

% card_Diff1_less
thf(fact_3630_card__Diff1__less,axiom,
    ! [A2: set_o,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A2 ) ) ) ) ).

% card_Diff1_less
thf(fact_3631_card__Diff1__less,axiom,
    ! [A2: set_int,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A2 ) ) ) ) ).

% card_Diff1_less
thf(fact_3632_card__Diff1__less,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).

% card_Diff1_less
thf(fact_3633_card__Diff__singleton__if,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
          = ( finite711546835091564841at_nat @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3634_card__Diff__singleton__if,axiom,
    ! [X: set_nat_rat,A2: set_set_nat_rat] :
      ( ( ( member_set_nat_rat @ X @ A2 )
       => ( ( finite8736671560171388117at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) ) )
          = ( minus_minus_nat @ ( finite8736671560171388117at_rat @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_set_nat_rat @ X @ A2 )
       => ( ( finite8736671560171388117at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) ) )
          = ( finite8736671560171388117at_rat @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3635_card__Diff__singleton__if,axiom,
    ! [X: complex,A2: set_complex] :
      ( ( ( member_complex @ X @ A2 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) )
          = ( minus_minus_nat @ ( finite_card_complex @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_complex @ X @ A2 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) )
          = ( finite_card_complex @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3636_card__Diff__singleton__if,axiom,
    ! [X: list_nat,A2: set_list_nat] :
      ( ( ( member_list_nat @ X @ A2 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_list_nat @ X @ A2 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
          = ( finite_card_list_nat @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3637_card__Diff__singleton__if,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ( ( member_set_nat @ X @ A2 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_set_nat @ X @ A2 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
          = ( finite_card_set_nat @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3638_card__Diff__singleton__if,axiom,
    ! [X: real,A2: set_real] :
      ( ( ( member_real @ X @ A2 )
       => ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( finite_card_real @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_real @ X @ A2 )
       => ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) )
          = ( finite_card_real @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3639_card__Diff__singleton__if,axiom,
    ! [X: $o,A2: set_o] :
      ( ( ( member_o @ X @ A2 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( finite_card_o @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_o @ X @ A2 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) )
          = ( finite_card_o @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3640_card__Diff__singleton__if,axiom,
    ! [X: int,A2: set_int] :
      ( ( ( member_int @ X @ A2 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_int @ X @ A2 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) )
          = ( finite_card_int @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3641_card__Diff__singleton__if,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( ( member_nat @ X @ A2 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_nat @ X @ A2 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
          = ( finite_card_nat @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3642_card__Diff__singleton,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3643_card__Diff__singleton,axiom,
    ! [X: set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ X @ A2 )
     => ( ( finite8736671560171388117at_rat @ ( minus_1626877696091177228at_rat @ A2 @ ( insert_set_nat_rat @ X @ bot_bo6797373522285170759at_rat ) ) )
        = ( minus_minus_nat @ ( finite8736671560171388117at_rat @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3644_card__Diff__singleton,axiom,
    ! [X: complex,A2: set_complex] :
      ( ( member_complex @ X @ A2 )
     => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) )
        = ( minus_minus_nat @ ( finite_card_complex @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3645_card__Diff__singleton,axiom,
    ! [X: list_nat,A2: set_list_nat] :
      ( ( member_list_nat @ X @ A2 )
     => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3646_card__Diff__singleton,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ X @ A2 )
     => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3647_card__Diff__singleton,axiom,
    ! [X: real,A2: set_real] :
      ( ( member_real @ X @ A2 )
     => ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) )
        = ( minus_minus_nat @ ( finite_card_real @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3648_card__Diff__singleton,axiom,
    ! [X: $o,A2: set_o] :
      ( ( member_o @ X @ A2 )
     => ( ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) )
        = ( minus_minus_nat @ ( finite_card_o @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3649_card__Diff__singleton,axiom,
    ! [X: int,A2: set_int] :
      ( ( member_int @ X @ A2 )
     => ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) )
        = ( minus_minus_nat @ ( finite_card_int @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3650_card__Diff__singleton,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( member_nat @ X @ A2 )
     => ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3651_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_3652_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_3653_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3654_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_3655_power__decreasing__iff,axiom,
    ! [B: real,M2: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3656_power__decreasing__iff,axiom,
    ! [B: rat,M2: 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 @ M2 ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3657_power__decreasing__iff,axiom,
    ! [B: nat,M2: 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 @ M2 ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3658_power__decreasing__iff,axiom,
    ! [B: int,M2: 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 @ M2 ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3659_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
      = ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_3660_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_3661_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_3662_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_3663_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_3664_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_3665_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_3666_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3667_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3668_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3669_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3670_power__strict__decreasing__iff,axiom,
    ! [B: real,M2: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3671_power__strict__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3672_power__strict__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3673_power__strict__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3674_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_3675_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3676_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_3677_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_3678_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_3679_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3680_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_3681_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_3682_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_3683_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_3684_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_3685_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3686_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_3687_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] :
          ( X
         != ( plus_plus_nat @ N2 @ N2 ) )
     => ~ ! [N2: nat] :
            ( X
           != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).

% even_odd_cases
thf(fact_3688_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_3689_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_3690_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_3691_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_real @ one_one_real @ N )
      = one_one_real ) ).

% power_one
thf(fact_3692_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ one_one_complex @ N )
      = one_one_complex ) ).

% power_one
thf(fact_3693_add__Suc__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc_right
thf(fact_3694_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_3695_add__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_3696_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_3697_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_3698_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_3699_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_3700_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_3701_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_3702_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_3703_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_3704_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_3705_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_3706_power__inject__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M2 )
          = ( power_power_real @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3707_power__inject__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M2 )
          = ( power_power_rat @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3708_power__inject__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M2 )
          = ( power_power_nat @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3709_power__inject__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M2 )
          = ( power_power_int @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3710_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_3711_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_3712_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_3713_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_3714_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_3715_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_3716_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_3717_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_3718_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_3719_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_3720_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_3721_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M2: nat] :
      ( ( ( power_power_nat @ X @ M2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M2 = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_3722_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_3723_mult__Suc__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( times_times_nat @ M2 @ ( suc @ N ) )
      = ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).

% mult_Suc_right
thf(fact_3724_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_3725_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_3726_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_3727_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_3728_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
          = ( divide_divide_nat @ M2 @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_3729_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3730_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3731_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3732_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3733_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_3734_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_3735_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_3736_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( power_power_real @ A @ N )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3737_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( power_power_complex @ A @ N )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3738_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_3739_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_3740_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_3741_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_3742_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3743_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_3744_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_3745_add__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc
thf(fact_3746_add__Suc__shift,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_3747_add__eq__self__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = M2 )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_3748_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_3749_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% less_add_eq_less
thf(fact_3750_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_3751_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_3752_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_3753_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_3754_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_3755_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_3756_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_3757_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).

% real_arch_pow
thf(fact_3758_add__leE,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M2 @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_3759_le__add1,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).

% le_add1
thf(fact_3760_le__add2,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).

% le_add2
thf(fact_3761_add__leD1,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% add_leD1
thf(fact_3762_add__leD2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_3763_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_3764_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_3765_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_3766_trans__le__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_le_add1
thf(fact_3767_trans__le__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_le_add2
thf(fact_3768_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_3769_Nat_Odiff__cancel,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% Nat.diff_cancel
thf(fact_3770_diff__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_cancel2
thf(fact_3771_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
      = M2 ) ).

% diff_add_inverse
thf(fact_3772_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
      = M2 ) ).

% diff_add_inverse2
thf(fact_3773_add__mult__distrib,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_3774_add__mult__distrib2,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_3775_nat__power__less__imp__less,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_3776_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_3777_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).

% nat_one_le_power
thf(fact_3778_add__is__1,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_3779_one__is__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M2 @ N ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_3780_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ~ ! [Q5: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M2 @ Q5 ) ) ) ) ).

% less_natE
thf(fact_3781_less__add__Suc1,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_3782_less__add__Suc2,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).

% less_add_Suc2
thf(fact_3783_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_3784_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_3785_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ one_one_real )
       => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_3786_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_3787_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_3788_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_3789_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_3790_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_3791_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M2
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_3792_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_3793_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M2: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( ( ord_less_nat @ M4 @ N2 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_3794_mult__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M2 ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ).

% mult_Suc
thf(fact_3795_diff__add__0,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_3796_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_3797_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_3798_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_3799_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_3800_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_3801_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_3802_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_3803_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_3804_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_3805_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_3806_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y4: real] :
        ? [N2: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_3807_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_3808_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_3809_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self
thf(fact_3810_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 ) )
            | ? [D5: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D5 ) )
                & ~ ( P @ D5 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_3811_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 ) )
        & ! [D5: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D5 ) )
           => ( P @ D5 ) ) ) ) ).

% nat_diff_split
thf(fact_3812_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_3813_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N4: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_3814_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_3815_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% add_eq_if
thf(fact_3816_dividend__less__times__div,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_3817_dividend__less__div__times,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_3818_split__div,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
               => ( P @ I4 ) ) ) ) ) ) ).

% split_div
thf(fact_3819_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).

% nat_less_real_le
thf(fact_3820_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_3821_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% mult_eq_if
thf(fact_3822_q__pos__lemma,axiom,
    ! [B7: int,Q6: int,R3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R3 ) )
     => ( ( ord_less_int @ R3 @ B7 )
       => ( ( ord_less_int @ zero_zero_int @ B7 )
         => ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).

% q_pos_lemma
thf(fact_3823_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B7: int,Q6: int,R3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R3 ) )
       => ( ( ord_less_int @ R3 @ B7 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B7 )
             => ( ( ord_less_eq_int @ B7 @ B )
               => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3824_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B7: int,Q6: int,R3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R3 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R3 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B7 )
             => ( ( ord_less_eq_int @ B7 @ B )
               => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3825_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R3: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( ord_less_int @ R2 @ B )
           => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3826_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R3: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( ord_less_int @ B @ R3 )
           => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3827_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int] :
            ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X2: int] :
              ( ( P @ X2 )
             => ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3828_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).

% nat_mult_distrib
thf(fact_3829_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).

% nat_power_eq
thf(fact_3830_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int] :
            ( ( P @ X4 )
           => ( P @ ( minus_minus_int @ X4 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X2: int] :
              ( ( P @ X2 )
             => ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3831_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_3832_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_3833_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M4 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_3834_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_3835_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( divide_divide_int @ A @ B )
            = Q4 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3836_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q4 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3837_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_3838_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_3839_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_3840_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_3841_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_3842_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M2 )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M2 = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_3843_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_3844_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_power
thf(fact_3845_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_3846_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_3847_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_3848_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono
thf(fact_3849_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_3850_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_3851_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_3852_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_less_power
thf(fact_3853_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_3854_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_3855_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_3856_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% one_le_power
thf(fact_3857_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_3858_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_3859_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_3860_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_3861_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_3862_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_3863_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_3864_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_3865_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_3866_power__one__over,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).

% power_one_over
thf(fact_3867_power__one__over,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% power_one_over
thf(fact_3868_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_3869_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_3870_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_3871_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_3872_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_3873_nat__mult__less__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_3874_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M2 )
          = ( times_times_nat @ K @ N ) )
        = ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_3875_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3876_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3877_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3878_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3879_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_3880_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_3881_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_3882_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3883_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3884_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3885_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_3886_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_3887_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_3888_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_3889_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3890_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_3891_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_3892_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_3893_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N ) )
        = ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3894_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_3895_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_3896_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_3897_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_3898_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_3899_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_3900_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_3901_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_3902_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_3903_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_3904_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_3905_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3906_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_3907_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_3908_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_3909_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_3910_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_3911_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_3912_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_3913_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_3914_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_3915_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_3916_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_3917_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_3918_power__less__imp__less__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3919_power__less__imp__less__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3920_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3921_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3922_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_3923_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_3924_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_3925_power__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_3926_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_3927_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_3928_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_3929_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_3930_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_3931_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_3932_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ zero_zero_real @ N )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_3933_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ zero_zero_complex @ N )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_3934_nat__mult__le__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_3935_nat__mult__div__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
        = ( divide_divide_nat @ M2 @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_3936_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_3937_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_3938_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_3939_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_3940_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3941_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_3942_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_3943_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_3944_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_3945_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_3946_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_3947_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_3948_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3949_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_3950_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_3951_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_3952_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3953_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_3954_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_3955_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_3956_power__le__imp__le__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3957_power__le__imp__le__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3958_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3959_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3960_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ N )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3961_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_3962_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_3963_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_3964_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N )
              = ( power_power_real @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3965_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_3966_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_3967_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_3968_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3969_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_3970_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_3971_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_3972_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_3973_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_3974_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_3975_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_3976_power__diff,axiom,
    ! [A: rat,N: nat,M2: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3977_power__diff,axiom,
    ! [A: complex,N: nat,M2: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3978_power__diff,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3979_power__diff,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3980_power__diff,axiom,
    ! [A: real,N: nat,M2: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3981_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3982_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_3983_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_3984_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_3985_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3986_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3987_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3988_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3989_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3990_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3991_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3992_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_3993_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_3994_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_3995_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_3996_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D6 )
               => ( ( ord_less_eq_real @ A @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_3997_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A5: real,B5: real,C3: real] :
            ( ( P @ A5 @ B5 )
           => ( ( P @ B5 @ C3 )
             => ( ( ord_less_eq_real @ A5 @ B5 )
               => ( ( ord_less_eq_real @ B5 @ C3 )
                 => ( P @ A5 @ C3 ) ) ) ) )
       => ( ! [X4: real] :
              ( ( ord_less_eq_real @ A @ X4 )
             => ( ( ord_less_eq_real @ X4 @ B )
               => ? [D3: real] :
                    ( ( ord_less_real @ zero_zero_real @ D3 )
                    & ! [A5: real,B5: real] :
                        ( ( ( ord_less_eq_real @ A5 @ X4 )
                          & ( ord_less_eq_real @ X4 @ B5 )
                          & ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D3 ) )
                       => ( P @ A5 @ B5 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_3998_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X4: real] :
            ( ( ord_less_real @ zero_zero_real @ X4 )
            & ( ( power_power_real @ X4 @ N )
              = A )
            & ! [Y4: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y4 )
                  & ( ( power_power_real @ Y4 @ N )
                    = A ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_3999_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ( ( power_power_real @ R4 @ N )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_4000_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4001_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4002_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4003_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4004_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4005_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4006_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4007_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4008_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4009_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4010_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_4011_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_4012_finite__maxlen,axiom,
    ! [M5: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M5 )
     => ? [N2: nat] :
        ! [X2: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X2 @ M5 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X2 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_4013_finite__maxlen,axiom,
    ! [M5: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M5 )
     => ? [N2: nat] :
        ! [X2: list_nat] :
          ( ( member_list_nat @ X2 @ M5 )
         => ( ord_less_nat @ ( size_size_list_nat @ X2 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_4014_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_4015_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_4016_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_4017_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_4018_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_4019_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_4020_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_4021_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_4022_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_4023_the__elem__eq,axiom,
    ! [X: real] :
      ( ( the_elem_real @ ( insert_real @ X @ bot_bot_set_real ) )
      = X ) ).

% the_elem_eq
thf(fact_4024_the__elem__eq,axiom,
    ! [X: $o] :
      ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% the_elem_eq
thf(fact_4025_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_4026_the__elem__eq,axiom,
    ! [X: int] :
      ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
      = X ) ).

% the_elem_eq
thf(fact_4027_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_4028_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).

% is_singletonI
thf(fact_4029_is__singletonI,axiom,
    ! [X: real] : ( is_singleton_real @ ( insert_real @ X @ bot_bot_set_real ) ) ).

% is_singletonI
thf(fact_4030_is__singletonI,axiom,
    ! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_4031_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_4032_is__singletonI,axiom,
    ! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).

% is_singletonI
thf(fact_4033_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( P @ A5 @ B5 )
          = ( P @ B5 @ A5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
       => ( ! [A5: nat,B5: nat] :
              ( ( P @ A5 @ B5 )
             => ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_4034_ln__root,axiom,
    ! [N: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% ln_root
thf(fact_4035_log__of__power__le,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_4036_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_4037_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_4038_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_4039_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_4040_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_4041_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_4042_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_4043_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_4044_gbinomial__0_I1_J,axiom,
    ! [A: complex] :
      ( ( gbinomial_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_4045_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_4046_gbinomial__0_I1_J,axiom,
    ! [A: rat] :
      ( ( gbinomial_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_4047_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_4048_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_4049_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = ( root @ N @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_4050_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_4051_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_arctan_iff
thf(fact_4052_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_4053_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_4054_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_4055_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_4056_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_4057_real__root__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_4058_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_4059_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_4060_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_4061_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_4062_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_4063_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_4064_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_4065_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_4066_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_4067_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_4068_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ A @ X ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_4069_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_eq_real @ X @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_4070_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_4071_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_4072_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone'
thf(fact_4073_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% arctan_le_iff
thf(fact_4074_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).

% real_root_pos_pos_le
thf(fact_4075_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_4076_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_4077_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_4078_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_4079_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_4080_is__singleton__the__elem,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
          ( A6
          = ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A6 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_4081_is__singleton__the__elem,axiom,
    ( is_singleton_real
    = ( ^ [A6: set_real] :
          ( A6
          = ( insert_real @ ( the_elem_real @ A6 ) @ bot_bot_set_real ) ) ) ) ).

% is_singleton_the_elem
thf(fact_4082_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
          ( A6
          = ( insert_o @ ( the_elem_o @ A6 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_4083_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
          ( A6
          = ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_4084_is__singleton__the__elem,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
          ( A6
          = ( insert_int @ ( the_elem_int @ A6 ) @ bot_bot_set_int ) ) ) ) ).

% is_singleton_the_elem
thf(fact_4085_is__singletonI_H,axiom,
    ! [A2: set_set_nat] :
      ( ( A2 != bot_bot_set_set_nat )
     => ( ! [X4: set_nat,Y3: set_nat] :
            ( ( member_set_nat @ X4 @ A2 )
           => ( ( member_set_nat @ Y3 @ A2 )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_set_nat @ A2 ) ) ) ).

% is_singletonI'
thf(fact_4086_is__singletonI_H,axiom,
    ! [A2: set_set_nat_rat] :
      ( ( A2 != bot_bo6797373522285170759at_rat )
     => ( ! [X4: set_nat_rat,Y3: set_nat_rat] :
            ( ( member_set_nat_rat @ X4 @ A2 )
           => ( ( member_set_nat_rat @ Y3 @ A2 )
             => ( X4 = Y3 ) ) )
       => ( is_sin2571591796506819849at_rat @ A2 ) ) ) ).

% is_singletonI'
thf(fact_4087_is__singletonI_H,axiom,
    ! [A2: set_real] :
      ( ( A2 != bot_bot_set_real )
     => ( ! [X4: real,Y3: real] :
            ( ( member_real @ X4 @ A2 )
           => ( ( member_real @ Y3 @ A2 )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_real @ A2 ) ) ) ).

% is_singletonI'
thf(fact_4088_is__singletonI_H,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ! [X4: $o,Y3: $o] :
            ( ( member_o @ X4 @ A2 )
           => ( ( member_o @ Y3 @ A2 )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_o @ A2 ) ) ) ).

% is_singletonI'
thf(fact_4089_is__singletonI_H,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X4: nat,Y3: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y3 @ A2 )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_nat @ A2 ) ) ) ).

% is_singletonI'
thf(fact_4090_is__singletonI_H,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
     => ( ! [X4: int,Y3: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ( member_int @ Y3 @ A2 )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_int @ A2 ) ) ) ).

% is_singletonI'
thf(fact_4091_log__root,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_root
thf(fact_4092_log__base__root,axiom,
    ! [N: nat,B: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N @ B ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).

% log_base_root
thf(fact_4093_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_4094_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_4095_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_4096_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_4097_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_4098_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_4099_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_4100_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_4101_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_4102_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_4103_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_4104_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).

% real_root_abs
thf(fact_4105_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_4106_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
      = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_4107_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_4108_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_4109_gbinomial__absorption,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_4110_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_4111_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M2: nat,A: real] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( times_times_real @ ( gbinomial_real @ A @ M2 ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M2 ) @ K ) )
        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M2 @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_4112_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M2: nat,A: rat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M2 ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M2 ) @ K ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M2 @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_4113_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_4114_real__root__strict__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_4115_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_4116_gbinomial__rec,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_4117_gbinomial__rec,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_4118_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_4119_gbinomial__factors,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_4120_gbinomial__factors,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_4121_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_4122_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_4123_real__root__strict__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_4124_real__root__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_4125_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_4126_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N )
            = X )
         => ( ( root @ N @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_4127_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N @ ( power_power_real @ X @ N ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_4128_le__log__of__power,axiom,
    ! [B: real,N: nat,M2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M2 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M2 ) ) ) ) ).

% le_log_of_power
thf(fact_4129_is__singletonE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( is_sin2850979758926227957at_nat @ A2 )
     => ~ ! [X4: product_prod_nat_nat] :
            ( A2
           != ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ).

% is_singletonE
thf(fact_4130_is__singletonE,axiom,
    ! [A2: set_real] :
      ( ( is_singleton_real @ A2 )
     => ~ ! [X4: real] :
            ( A2
           != ( insert_real @ X4 @ bot_bot_set_real ) ) ) ).

% is_singletonE
thf(fact_4131_is__singletonE,axiom,
    ! [A2: set_o] :
      ( ( is_singleton_o @ A2 )
     => ~ ! [X4: $o] :
            ( A2
           != ( insert_o @ X4 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_4132_is__singletonE,axiom,
    ! [A2: set_nat] :
      ( ( is_singleton_nat @ A2 )
     => ~ ! [X4: nat] :
            ( A2
           != ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_4133_is__singletonE,axiom,
    ! [A2: set_int] :
      ( ( is_singleton_int @ A2 )
     => ~ ! [X4: int] :
            ( A2
           != ( insert_int @ X4 @ bot_bot_set_int ) ) ) ).

% is_singletonE
thf(fact_4134_is__singleton__def,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
        ? [X3: product_prod_nat_nat] :
          ( A6
          = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_4135_is__singleton__def,axiom,
    ( is_singleton_real
    = ( ^ [A6: set_real] :
        ? [X3: real] :
          ( A6
          = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).

% is_singleton_def
thf(fact_4136_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
        ? [X3: $o] :
          ( A6
          = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_4137_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
        ? [X3: nat] :
          ( A6
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_4138_is__singleton__def,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
        ? [X3: int] :
          ( A6
          = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).

% is_singleton_def
thf(fact_4139_is__singleton__altdef,axiom,
    ( is_singleton_complex
    = ( ^ [A6: set_complex] :
          ( ( finite_card_complex @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_4140_is__singleton__altdef,axiom,
    ( is_sin2641923865335537900st_nat
    = ( ^ [A6: set_list_nat] :
          ( ( finite_card_list_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_4141_is__singleton__altdef,axiom,
    ( is_singleton_set_nat
    = ( ^ [A6: set_set_nat] :
          ( ( finite_card_set_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_4142_is__singleton__altdef,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
          ( ( finite_card_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_4143_is__singleton__altdef,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
          ( ( finite_card_int @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_4144_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_4145_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_4146_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_4147_real__root__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_4148_log__of__power__less,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_4149_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_4150_gbinomial__minus,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_4151_gbinomial__minus,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_4152_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_4153_split__root,axiom,
    ! [P: real > $o,N: nat,X: real] :
      ( ( P @ ( root @ N @ X ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N )
         => ! [Y2: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
                = X )
             => ( P @ Y2 ) ) ) ) ) ).

% split_root
thf(fact_4154_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_4155_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_4156_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_4157_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_4158_Gcd__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( ( gcd_Gcd_nat @ A2 )
        = zero_zero_nat )
      = ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% Gcd_0_iff
thf(fact_4159_Gcd__0__iff,axiom,
    ! [A2: set_int] :
      ( ( ( gcd_Gcd_int @ A2 )
        = zero_zero_int )
      = ( ord_less_eq_set_int @ A2 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).

% Gcd_0_iff
thf(fact_4160_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_4161_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4162_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_4163_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4164_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4165_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4166_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4167_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4168_Compl__anti__mono,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_4169_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B2 ) )
      = ( ord_less_eq_set_int @ B2 @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_4170_sgn__sgn,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
      = ( sgn_sgn_real @ A ) ) ).

% sgn_sgn
thf(fact_4171_sgn__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
      = ( sgn_sgn_int @ A ) ) ).

% sgn_sgn
thf(fact_4172_sgn__sgn,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
      = ( sgn_sgn_complex @ A ) ) ).

% sgn_sgn
thf(fact_4173_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4174_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_4175_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_4176_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
      = ( ord_less_eq_set_int @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_4177_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_4178_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_4179_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_4180_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_4181_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_4182_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A ) )
      = ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4183_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_4184_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4185_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_4186_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_4187_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_4188_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_4189_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_4190_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_4191_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_4192_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_4193_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_4194_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_4195_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_4196_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4197_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_4198_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_4199_mult__minus__right,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4200_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_4201_mult__minus__right,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4202_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_4203_minus__mult__minus,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( times_times_real @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4204_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_4205_minus__mult__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( times_times_complex @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4206_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_4207_mult__minus__left,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4208_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_4209_mult__minus__left,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4210_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_4211_minus__add__distrib,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).

% minus_add_distrib
thf(fact_4212_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_4213_minus__add__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_add_distrib
thf(fact_4214_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_4215_minus__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4216_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_4217_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4218_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_4219_add__minus__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4220_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_4221_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4222_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_4223_minus__diff__eq,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
      = ( minus_minus_real @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4224_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_4225_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4226_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_4227_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_4228_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_4229_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_4230_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_4231_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_4232_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_4233_sgn__0,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_0
thf(fact_4234_sgn__0,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_0
thf(fact_4235_sgn__0,axiom,
    ( ( sgn_sgn_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% sgn_0
thf(fact_4236_sgn__0,axiom,
    ( ( sgn_sgn_int @ zero_zero_int )
    = zero_zero_int ) ).

% sgn_0
thf(fact_4237_powr__0,axiom,
    ! [Z: real] :
      ( ( powr_real @ zero_zero_real @ Z )
      = zero_zero_real ) ).

% powr_0
thf(fact_4238_powr__eq__0__iff,axiom,
    ! [W2: real,Z: real] :
      ( ( ( powr_real @ W2 @ Z )
        = zero_zero_real )
      = ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
thf(fact_4239_sgn__1,axiom,
    ( ( sgn_sgn_rat @ one_one_rat )
    = one_one_rat ) ).

% sgn_1
thf(fact_4240_sgn__1,axiom,
    ( ( sgn_sgn_real @ one_one_real )
    = one_one_real ) ).

% sgn_1
thf(fact_4241_sgn__1,axiom,
    ( ( sgn_sgn_int @ one_one_int )
    = one_one_int ) ).

% sgn_1
thf(fact_4242_sgn__1,axiom,
    ( ( sgn_sgn_complex @ one_one_complex )
    = one_one_complex ) ).

% sgn_1
thf(fact_4243_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
      = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_4244_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
      = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_4245_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
      = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_4246_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_4247_powr__one__eq__one,axiom,
    ! [A: real] :
      ( ( powr_real @ one_one_real @ A )
      = one_one_real ) ).

% powr_one_eq_one
thf(fact_4248_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4249_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_4250_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_4251_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_4252_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_4253_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_4254_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4255_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_4256_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_4257_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_4258_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_4259_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_4260_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_4261_less__neg__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% less_neg_neg
thf(fact_4262_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_4263_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_4264_neg__less__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_pos
thf(fact_4265_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_4266_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_4267_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% neg_0_less_iff_less
thf(fact_4268_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_4269_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_4270_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4271_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_4272_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_4273_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_4274_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_4275_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_4276_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_4277_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_4278_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_4279_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_4280_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_4281_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_4282_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_4283_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_4284_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_4285_verit__minus__simplify_I3_J,axiom,
    ! [B: real] :
      ( ( minus_minus_real @ zero_zero_real @ B )
      = ( uminus_uminus_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4286_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_4287_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4288_mult__minus1__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1_right
thf(fact_4289_mult__minus1__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1_right
thf(fact_4290_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_4291_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_4292_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_4293_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_4294_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_4295_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_4296_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_4297_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
      = ( minus_minus_real @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4298_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_4299_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( minus_minus_complex @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4300_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_4301_diff__minus__eq__add,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
      = ( plus_plus_real @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4302_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_4303_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( plus_plus_complex @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4304_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_4305_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_4306_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_4307_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_4308_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_4309_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_4310_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_4311_sgn__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% sgn_less
thf(fact_4312_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_4313_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_4314_sgn__greater,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% sgn_greater
thf(fact_4315_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_4316_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_4317_subset__Compl__singleton,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4318_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4319_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat_rat,B: set_nat_rat] :
      ( ( ord_le4375437777232675859at_rat @ A2 @ ( uminus3098529973357106300at_rat @ ( insert_set_nat_rat @ B @ bot_bo6797373522285170759at_rat ) ) )
      = ( ~ ( member_set_nat_rat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4320_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4321_subset__Compl__singleton,axiom,
    ! [A2: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
      = ( ~ ( member_o @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4322_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4323_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4324_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( ( powr_real @ X @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( powr_real @ X @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_4325_negative__eq__positive,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M2 ) )
      = ( ( N = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_4326_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X = A ) ) ).

% real_add_minus_iff
thf(fact_4327_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_4328_Gcd__empty,axiom,
    ( ( gcd_Gcd_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_empty
thf(fact_4329_Gcd__empty,axiom,
    ( ( gcd_Gcd_int @ bot_bot_set_int )
    = zero_zero_int ) ).

% Gcd_empty
thf(fact_4330_negative__zle,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).

% negative_zle
thf(fact_4331_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_4332_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_inc_simps(4)
thf(fact_4333_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_4334_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_4335_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_4336_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = zero_zero_real ) ).

% add_neg_numeral_special(8)
thf(fact_4337_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_4338_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = zero_zero_complex ) ).

% add_neg_numeral_special(8)
thf(fact_4339_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_4340_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% add_neg_numeral_special(7)
thf(fact_4341_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_4342_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% add_neg_numeral_special(7)
thf(fact_4343_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_4344_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% diff_numeral_special(12)
thf(fact_4345_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_4346_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% diff_numeral_special(12)
thf(fact_4347_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_4348_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4349_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_4350_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4351_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_4352_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_4353_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_4354_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_4355_abs__of__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_nonpos
thf(fact_4356_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_4357_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_4358_sgn__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( sgn_sgn_real @ A )
        = one_one_real ) ) ).

% sgn_pos
thf(fact_4359_sgn__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( sgn_sgn_rat @ A )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_4360_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_4361_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
        = one_one_real ) ) ).

% abs_sgn_eq_1
thf(fact_4362_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_4363_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_4364_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_4365_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ( powr_real @ X @ one_one_real )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% powr_one_gt_zero_iff
thf(fact_4366_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_4367_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_4368_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_4369_sgn__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( sgn_sgn_real @ A )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% sgn_neg
thf(fact_4370_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_4371_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4372_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4373_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4374_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4375_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_4376_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_4377_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_4378_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_4379_sgn__minus__1,axiom,
    ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sgn_minus_1
thf(fact_4380_sgn__minus__1,axiom,
    ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sgn_minus_1
thf(fact_4381_sgn__minus__1,axiom,
    ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% sgn_minus_1
thf(fact_4382_sgn__minus__1,axiom,
    ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% sgn_minus_1
thf(fact_4383_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_4384_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
       != ( sgn_sgn_real @ A ) )
     => ( ( ( sgn_sgn_real @ A )
         != zero_zero_real )
       => ( ( ( sgn_sgn_real @ B )
           != zero_zero_real )
         => ( ( sgn_sgn_real @ A )
            = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_4385_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_4386_sgn__0__0,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_0_0
thf(fact_4387_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% sgn_0_0
thf(fact_4388_sgn__0__0,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_0_0
thf(fact_4389_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( sgn_sgn_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
thf(fact_4390_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
thf(fact_4391_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
thf(fact_4392_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
thf(fact_4393_sgn__mult,axiom,
    ! [A: complex,B: complex] :
      ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B ) )
      = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).

% sgn_mult
thf(fact_4394_sgn__mult,axiom,
    ! [A: real,B: real] :
      ( ( sgn_sgn_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).

% sgn_mult
thf(fact_4395_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_4396_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_4397_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
        = ( sgn_sgn_real @ A ) )
     => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
        = ( sgn_sgn_real @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_4398_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_4399_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_4400_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_4401_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_4402_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_4403_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_4404_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_4405_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_4406_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_4407_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_4408_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_4409_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).

% compl_mono
thf(fact_4410_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
     => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% compl_le_swap1
thf(fact_4411_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_4412_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_4413_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% less_minus_iff
thf(fact_4414_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_4415_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_4416_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_4417_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_4418_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_4419_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_4420_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_4421_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_4422_minus__mult__commute,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_mult_commute
thf(fact_4423_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_4424_minus__mult__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_mult_commute
thf(fact_4425_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_4426_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% square_eq_iff
thf(fact_4427_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_4428_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% square_eq_iff
thf(fact_4429_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_4430_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_4431_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_4432_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_4433_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_4434_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_4435_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_4436_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_4437_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_4438_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_4439_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_4440_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2
        = ( plus_plus_complex @ K @ A ) )
     => ( ( uminus1482373934393186551omplex @ A2 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_4441_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_4442_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_4443_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_4444_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_4445_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_4446_minus__diff__commute,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_4447_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_4448_minus__diff__commute,axiom,
    ! [B: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_4449_minus__diff__minus,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_4450_minus__diff__minus,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_4451_minus__diff__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_4452_minus__diff__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_4453_powr__minus__divide,axiom,
    ! [X: real,A: real] :
      ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
      = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).

% powr_minus_divide
thf(fact_4454_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( abs_abs_int @ X )
        = ( abs_abs_int @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_4455_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( abs_abs_real @ X )
        = ( abs_abs_real @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_4456_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( abs_abs_rat @ X )
        = ( abs_abs_rat @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_4457_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_4458_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_4459_sgn__if,axiom,
    ( sgn_sgn_real
    = ( ^ [X3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_if
thf(fact_4460_sgn__if,axiom,
    ( sgn_sgn_rat
    = ( ^ [X3: rat] : ( if_rat @ ( X3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_if
thf(fact_4461_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_4462_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% sgn_1_neg
thf(fact_4463_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_4464_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_4465_Gcd__1,axiom,
    ! [A2: set_nat] :
      ( ( member_nat @ one_one_nat @ A2 )
     => ( ( gcd_Gcd_nat @ A2 )
        = one_one_nat ) ) ).

% Gcd_1
thf(fact_4466_Gcd__1,axiom,
    ! [A2: set_int] :
      ( ( member_int @ one_one_int @ A2 )
     => ( ( gcd_Gcd_int @ A2 )
        = one_one_int ) ) ).

% Gcd_1
thf(fact_4467_powr__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_4468_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).

% powr_ge_pzero
thf(fact_4469_powr__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X )
       => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_mono
thf(fact_4470_mult__sgn__abs,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ X ) @ ( abs_abs_real @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_4471_mult__sgn__abs,axiom,
    ! [X: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_4472_mult__sgn__abs,axiom,
    ! [X: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_4473_sgn__mult__abs,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_4474_sgn__mult__abs,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_4475_sgn__mult__abs,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_4476_sgn__mult__abs,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_4477_abs__mult__sgn,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_4478_abs__mult__sgn,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_4479_abs__mult__sgn,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_4480_abs__mult__sgn,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_4481_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_real
    = ( ^ [K3: real] : ( times_times_real @ K3 @ ( sgn_sgn_real @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_4482_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_4483_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_4484_same__sgn__abs__add,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
        = ( sgn_sgn_real @ A ) )
     => ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_4485_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_4486_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_4487_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% le_minus_one_simps(4)
thf(fact_4488_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_4489_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_4490_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% le_minus_one_simps(2)
thf(fact_4491_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_4492_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_4493_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_4494_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4495_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_4496_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4497_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_4498_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_4499_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_4500_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_group_add_class.ab_left_minus
thf(fact_4501_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_4502_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_4503_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_4504_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_4505_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_4506_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4507_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_4508_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4509_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_4510_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4511_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_4512_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4513_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_4514_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_4515_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_4516_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_4517_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_4518_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(4)
thf(fact_4519_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_4520_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_4521_less__minus__one__simps_I2_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% less_minus_one_simps(2)
thf(fact_4522_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_4523_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_4524_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_4525_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_4526_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_4527_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_4528_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_4529_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_4530_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( times_times_real @ X @ X )
        = one_one_real )
      = ( ( X = one_one_real )
        | ( X
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_4531_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_4532_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( times_times_complex @ X @ X )
        = one_one_complex )
      = ( ( X = one_one_complex )
        | ( X
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_4533_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_4534_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_4535_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_4536_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,A: complex] :
      ( ( B2
        = ( plus_plus_complex @ K @ B ) )
     => ( ( minus_minus_complex @ A @ B2 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_4537_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4538_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4539_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4540_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4541_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4542_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4543_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4544_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4545_abs__ge__minus__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).

% abs_ge_minus_self
thf(fact_4546_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_4547_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_4548_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_eq_real @ A @ B )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_4549_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_4550_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_4551_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_4552_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_4553_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_4554_abs__leI,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
       => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_4555_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_4556_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_4557_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_4558_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_real @ A @ B )
        & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_4559_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_4560_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
thf(fact_4561_subset__Compl__self__eq,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_4562_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_4563_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_4564_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).

% real_minus_mult_self_le
thf(fact_4565_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X3: real,Y2: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).

% minus_real_def
thf(fact_4566_Gcd__int__greater__eq__0,axiom,
    ! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K4 ) ) ).

% Gcd_int_greater_eq_0
thf(fact_4567_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_4568_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_4569_powr__le1,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_4570_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ( ord_less_eq_real @ X @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_4571_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_4572_powr__divide,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
          = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_divide
thf(fact_4573_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_4574_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = one_one_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% sgn_1_pos
thf(fact_4575_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_4576_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_4577_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( sgn_sgn_real @ ( root @ N @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_4578_abs__sgn__eq,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
          = one_one_real ) ) ) ).

% abs_sgn_eq
thf(fact_4579_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_4580_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_4581_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% le_minus_one_simps(3)
thf(fact_4582_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_4583_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_4584_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% le_minus_one_simps(1)
thf(fact_4585_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_4586_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_4587_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_4588_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(3)
thf(fact_4589_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_4590_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_4591_less__minus__one__simps_I1_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% less_minus_one_simps(1)
thf(fact_4592_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_4593_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = ( uminus_uminus_real @ B ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4594_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_4595_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = ( uminus1482373934393186551omplex @ B ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4596_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B )
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4597_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_4598_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B )
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4599_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
          = C )
        = ( ( uminus_uminus_real @ A )
          = ( times_times_real @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4600_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_4601_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A )
          = ( times_times_complex @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4602_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
        = ( ( times_times_real @ C @ B )
          = ( uminus_uminus_real @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4603_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_4604_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ( times_times_complex @ C @ B )
          = ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4605_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B != zero_zero_real )
        & ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_4606_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_4607_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B != zero_zero_complex )
        & ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_4608_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_4609_power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).

% power_minus
thf(fact_4610_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_4611_power__minus,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_minus
thf(fact_4612_abs__minus__le__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).

% abs_minus_le_zero
thf(fact_4613_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_4614_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_4615_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( ( abs_abs_real @ A )
        = B )
      = ( ( ord_less_eq_real @ zero_zero_real @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_real @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_4616_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_4617_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_4618_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( abs_abs_real @ B ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_real @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_4619_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_4620_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_4621_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_4622_abs__of__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_neg
thf(fact_4623_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_4624_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_4625_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_4626_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_4627_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_4628_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_4629_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_4630_Compl__insert,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) )
      = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A2 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).

% Compl_insert
thf(fact_4631_Compl__insert,axiom,
    ! [X: real,A2: set_real] :
      ( ( uminus612125837232591019t_real @ ( insert_real @ X @ A2 ) )
      = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% Compl_insert
thf(fact_4632_Compl__insert,axiom,
    ! [X: $o,A2: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o @ X @ A2 ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_4633_Compl__insert,axiom,
    ! [X: int,A2: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_4634_Compl__insert,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_4635_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N2: nat] :
          ( M2
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( M2
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% int_cases4
thf(fact_4636_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_less_0_iff
thf(fact_4637_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_less_add_iff
thf(fact_4638_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_le_0_iff
thf(fact_4639_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_le_add_iff
thf(fact_4640_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_real_def
thf(fact_4641_int__zle__neg,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
      = ( ( N = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_4642_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% nonpos_int_cases
thf(fact_4643_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_4644_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( gcd_Gcd_nat @ M5 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M5 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_4645_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_4646_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_4647_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_4648_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_4649_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_4650_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_4651_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4652_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_4653_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4654_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_4655_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4656_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_4657_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4658_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4659_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4660_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4661_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4662_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4663_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4664_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4665_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4666_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4667_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4668_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4669_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4670_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4671_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4672_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
       => ~ ! [N2: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% int_cases3
thf(fact_4673_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_4674_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,N: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_4675_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
        = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_4676_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_4677_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% le_log_iff
thf(fact_4678_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_le_iff
thf(fact_4679_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% le_powr_iff
thf(fact_4680_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_le_iff
thf(fact_4681_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4682_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_4683_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4684_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_4685_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4686_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_4687_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4688_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_4689_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4690_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_4691_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4692_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_4693_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% neg_int_cases
thf(fact_4694_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_4695_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self2
thf(fact_4696_frac__neg,axiom,
    ! [X: real] :
      ( ( ( member_real @ X @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
          = zero_zero_real ) )
      & ( ~ ( member_real @ X @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
          = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).

% frac_neg
thf(fact_4697_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_4698_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_4699_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).

% ln_powr_bound2
thf(fact_4700_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
        = X ) ) ).

% sgn_power_root
thf(fact_4701_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_4702_gbinomial__negated__upper,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A4 ) @ one_one_complex ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_4703_gbinomial__negated__upper,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A4 ) @ one_one_real ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_4704_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A4 ) @ one_one_rat ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_4705_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_4706_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_4707_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_4708_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_sgn_iff
thf(fact_4709_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_4710_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_4711_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_4712_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_4713_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4714_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_4715_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4716_sgn__one,axiom,
    ( ( sgn_sgn_real @ one_one_real )
    = one_one_real ) ).

% sgn_one
thf(fact_4717_sgn__one,axiom,
    ( ( sgn_sgn_complex @ one_one_complex )
    = one_one_complex ) ).

% sgn_one
thf(fact_4718_sgn__zero,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_zero
thf(fact_4719_sgn__zero,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_zero
thf(fact_4720_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_4721_Compl__iff,axiom,
    ! [C: $o,A2: set_o] :
      ( ( member_o @ C @ ( uminus_uminus_set_o @ A2 ) )
      = ( ~ ( member_o @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_4722_Compl__iff,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
      = ( ~ ( member_set_nat @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_4723_Compl__iff,axiom,
    ! [C: set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ C @ ( uminus3098529973357106300at_rat @ A2 ) )
      = ( ~ ( member_set_nat_rat @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_4724_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( ~ ( member_nat @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_4725_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( ~ ( member_int @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_4726_ComplI,axiom,
    ! [C: $o,A2: set_o] :
      ( ~ ( member_o @ C @ A2 )
     => ( member_o @ C @ ( uminus_uminus_set_o @ A2 ) ) ) ).

% ComplI
thf(fact_4727_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ ( member_set_nat @ C @ A2 )
     => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).

% ComplI
thf(fact_4728_ComplI,axiom,
    ! [C: set_nat_rat,A2: set_set_nat_rat] :
      ( ~ ( member_set_nat_rat @ C @ A2 )
     => ( member_set_nat_rat @ C @ ( uminus3098529973357106300at_rat @ A2 ) ) ) ).

% ComplI
thf(fact_4729_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ ( member_nat @ C @ A2 )
     => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).

% ComplI
thf(fact_4730_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ ( member_int @ C @ A2 )
     => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% ComplI
thf(fact_4731_ceiling__zero,axiom,
    ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4732_ceiling__zero,axiom,
    ( ( archim7802044766580827645g_real @ zero_zero_real )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4733_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_4734_ceiling__one,axiom,
    ( ( archim7802044766580827645g_real @ one_one_real )
    = one_one_int ) ).

% ceiling_one
thf(fact_4735_ceiling__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_4736_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_4737_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_4738_zero__less__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% zero_less_ceiling
thf(fact_4739_ceiling__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_4740_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_4741_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_4742_one__le__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_le_ceiling
thf(fact_4743_ceiling__le__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_4744_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_4745_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_4746_one__less__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ one_one_real @ X ) ) ).

% one_less_ceiling
thf(fact_4747_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_4748_ceiling__add__one,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_4749_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_4750_ceiling__diff__one,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_4751_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
      = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_4752_ceiling__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
      = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_4753_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_4754_zero__le__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).

% zero_le_ceiling
thf(fact_4755_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_4756_ComplD,axiom,
    ! [C: $o,A2: set_o] :
      ( ( member_o @ C @ ( uminus_uminus_set_o @ A2 ) )
     => ~ ( member_o @ C @ A2 ) ) ).

% ComplD
thf(fact_4757_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
     => ~ ( member_set_nat @ C @ A2 ) ) ).

% ComplD
thf(fact_4758_ComplD,axiom,
    ! [C: set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ C @ ( uminus3098529973357106300at_rat @ A2 ) )
     => ~ ( member_set_nat_rat @ C @ A2 ) ) ).

% ComplD
thf(fact_4759_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
     => ~ ( member_nat @ C @ A2 ) ) ).

% ComplD
thf(fact_4760_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
     => ~ ( member_int @ C @ A2 ) ) ).

% ComplD
thf(fact_4761_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).

% ceiling_mono
thf(fact_4762_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% ceiling_mono
thf(fact_4763_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_4764_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_4765_of__nat__ceiling,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).

% of_nat_ceiling
thf(fact_4766_of__nat__ceiling,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).

% of_nat_ceiling
thf(fact_4767_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).

% ceiling_add_le
thf(fact_4768_ceiling__add__le,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).

% ceiling_add_le
thf(fact_4769_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_4770_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_4771_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_4772_sgn__zero__iff,axiom,
    ! [X: complex] :
      ( ( ( sgn_sgn_complex @ X )
        = zero_zero_complex )
      = ( X = zero_zero_complex ) ) ).

% sgn_zero_iff
thf(fact_4773_sgn__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sgn_sgn_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% sgn_zero_iff
thf(fact_4774_powr__int,axiom,
    ! [X: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_4775_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_4776_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_4777_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_4778_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_4779_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_4780_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_4781_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( power_power_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_4782_pochhammer__minus,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_4783_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_4784_pochhammer__minus,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_4785_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_4786_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4787_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_4788_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4789_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_4790_ceiling__eq,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_4791_ceiling__eq,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_4792_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_4793_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_4794_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% exp_le_cancel_iff
thf(fact_4795_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
        = X )
      = ( ? [N4: int] :
            ( X
            = ( ring_1_of_int_rat @ N4 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_4796_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
        = X )
      = ( ? [N4: int] :
            ( X
            = ( ring_1_of_int_real @ N4 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_4797_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_4798_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_4799_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_4800_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_4801_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_4802_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_4803_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_4804_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4805_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real
        = ( ring_1_of_int_real @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4806_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat
        = ( ring_1_of_int_rat @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4807_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = zero_zero_int )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4808_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = zero_zero_real )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4809_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = zero_zero_rat )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4810_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_4811_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_4812_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% of_int_le_iff
thf(fact_4813_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% of_int_le_iff
thf(fact_4814_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% of_int_le_iff
thf(fact_4815_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% of_int_less_iff
thf(fact_4816_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% of_int_less_iff
thf(fact_4817_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% of_int_less_iff
thf(fact_4818_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_4819_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_4820_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_4821_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_4822_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = one_one_complex )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4823_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = one_one_int )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4824_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = one_one_real )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4825_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = one_one_rat )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4826_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_4827_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_4828_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_4829_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_4830_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_4831_frac__of__int,axiom,
    ! [Z: int] :
      ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
      = zero_zero_real ) ).

% frac_of_int
thf(fact_4832_frac__of__int,axiom,
    ! [Z: int] :
      ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
      = zero_zero_rat ) ).

% frac_of_int
thf(fact_4833_exp__le__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_4834_one__le__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% one_le_exp_iff
thf(fact_4835_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).

% lesseq_shift
thf(fact_4836_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_4837_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_4838_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_4839_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4840_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4841_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4842_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_4843_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_4844_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_4845_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4846_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4847_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4848_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_4849_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_4850_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_4851_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_4852_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_4853_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_4854_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_4855_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_4856_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_4857_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_4858_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_4859_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_4860_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4861_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_4862_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_4863_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4864_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_4865_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_4866_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4867_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_4868_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_4869_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4870_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_4871_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_4872_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = ( ring_1_of_int_int @ Z ) ) ) ).

% of_nat_nat
thf(fact_4873_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
        = ( ring_1_of_int_real @ Z ) ) ) ).

% of_nat_nat
thf(fact_4874_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
        = ( ring_1_of_int_rat @ Z ) ) ) ).

% of_nat_nat
thf(fact_4875_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_le_of_int
thf(fact_4876_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_4877_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_less_of_int
thf(fact_4878_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_4879_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).

% ex_of_int_less
thf(fact_4880_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_4881_exp__not__eq__zero,axiom,
    ! [X: real] :
      ( ( exp_real @ X )
     != zero_zero_real ) ).

% exp_not_eq_zero
thf(fact_4882_exp__ge__zero,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_ge_zero
thf(fact_4883_not__exp__le__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_4884_le__of__int__ceiling,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_4885_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_4886_pochhammer__pos,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_4887_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_4888_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_4889_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_4890_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s7457072308508201937r_real @ A @ M2 )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_4891_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s4028243227959126397er_rat @ A @ M2 )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_4892_pochhammer__neq__0__mono,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M2 )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s7457072308508201937r_real @ A @ N )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_4893_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M2 )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s4028243227959126397er_rat @ A @ N )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_4894_exp__ge__add__one__self,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).

% exp_ge_add_one_self
thf(fact_4895_exp__minus__inverse,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
      = one_one_real ) ).

% exp_minus_inverse
thf(fact_4896_exp__minus__inverse,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
      = one_one_complex ) ).

% exp_minus_inverse
thf(fact_4897_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).

% ceiling_le
thf(fact_4898_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).

% ceiling_le
thf(fact_4899_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
      = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_4900_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_4901_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_4902_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_4903_pochhammer__nonneg,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_4904_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_4905_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_4906_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_4907_real__of__int__div4,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).

% real_of_int_div4
thf(fact_4908_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% pochhammer_0_left
thf(fact_4909_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_4910_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_4911_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_4912_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_4913_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_4914_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_nonneg
thf(fact_4915_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_nonneg
thf(fact_4916_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_nonneg
thf(fact_4917_of__int__leD,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% of_int_leD
thf(fact_4918_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_4919_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_4920_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_pos
thf(fact_4921_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_pos
thf(fact_4922_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_pos
thf(fact_4923_of__int__lessD,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% of_int_lessD
thf(fact_4924_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_4925_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_4926_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
          & ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X4 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_4927_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).

% ln_ge_iff
thf(fact_4928_floor__exists,axiom,
    ! [X: real] :
    ? [Z3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_4929_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_4930_floor__exists1,axiom,
    ! [X: real] :
    ? [X4: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
thf(fact_4931_floor__exists1,axiom,
    ! [X: rat] :
    ? [X4: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
thf(fact_4932_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).

% of_int_ceiling_le_add_one
thf(fact_4933_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_4934_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_4935_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_4936_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_4937_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_4938_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_4939_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_4940_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N4: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N4 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_4941_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N4: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N4 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).

% int_less_real_le
thf(fact_4942_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] :
      ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% ceiling_divide_eq_div
thf(fact_4943_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% ceiling_divide_eq_div
thf(fact_4944_pochhammer__rec,axiom,
    ! [A: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_4945_pochhammer__rec,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_4946_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_4947_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_4948_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_4949_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_4950_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_4951_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_4952_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_4953_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
        = zero_zero_complex )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_4954_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_4955_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
        = zero_zero_real )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_4956_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_4957_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N )
        = zero_zero_complex )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_4958_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_4959_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_4960_powr__def,axiom,
    ( powr_real
    = ( ^ [X3: real,A4: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A4 @ ( ln_ln_real @ X3 ) ) ) ) ) ) ).

% powr_def
thf(fact_4961_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_4962_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_4963_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_4964_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_4965_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_4966_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_4967_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( ( archim7802044766580827645g_real @ X )
        = A )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
        & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_4968_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_4969_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
       => ( ( archim7802044766580827645g_real @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_4970_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_4971_ceiling__correct,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
      & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% ceiling_correct
thf(fact_4972_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_4973_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_4974_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_4975_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% le_ceiling_iff
thf(fact_4976_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).

% le_ceiling_iff
thf(fact_4977_exp__divide__power__eq,axiom,
    ! [N: nat,X: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
        = ( exp_complex @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_4978_exp__divide__power__eq,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
        = ( exp_real @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_4979_real__of__int__div2,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).

% real_of_int_div2
thf(fact_4980_real__of__int__div3,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_4981_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_4982_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4660882817536571857er_int @ Z @ N )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_4983_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s7457072308508201937r_real @ Z @ N )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_4984_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_4985_ceiling__divide__upper,axiom,
    ! [Q4: real,P6: real] :
      ( ( ord_less_real @ zero_zero_real @ Q4 )
     => ( ord_less_eq_real @ P6 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q4 ) ) ) @ Q4 ) ) ) ).

% ceiling_divide_upper
thf(fact_4986_ceiling__divide__upper,axiom,
    ! [Q4: rat,P6: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q4 )
     => ( ord_less_eq_rat @ P6 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q4 ) ) ) @ Q4 ) ) ) ).

% ceiling_divide_upper
thf(fact_4987_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_4988_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_4989_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_4990_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_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_4991_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_4992_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_4993_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X3: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_4994_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_4995_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_4996_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_4997_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
      = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4998_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
      = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4999_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
      = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_5000_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
      = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_5001_ceiling__divide__lower,axiom,
    ! [Q4: rat,P6: 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 @ P6 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) @ P6 ) ) ).

% ceiling_divide_lower
thf(fact_5002_ceiling__divide__lower,axiom,
    ! [Q4: real,P6: real] :
      ( ( ord_less_real @ zero_zero_real @ Q4 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) @ P6 ) ) ).

% ceiling_divide_lower
thf(fact_5003_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y2: nat,X3: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).

% greater_shift
thf(fact_5004_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).

% less_shift
thf(fact_5005_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_correct
thf(fact_5006_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% pred_correct
thf(fact_5007_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_corr
thf(fact_5008_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% pred_corr
thf(fact_5009_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) ) ) ) ).

% maxt_sound
thf(fact_5010_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_5011_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_5012_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_5013_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% maxt_corr
thf(fact_5014_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% mint_corr
thf(fact_5015_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) ) ) ) ).

% mint_sound
thf(fact_5016_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_5017_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).

% mint_corr_help
thf(fact_5018_option_Osize_I4_J,axiom,
    ! [X23: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X23 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_5019_option_Osize_I4_J,axiom,
    ! [X23: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X23 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_5020_option_Osize_I4_J,axiom,
    ! [X23: num] :
      ( ( size_size_option_num @ ( some_num @ X23 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_5021_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_5022_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_5023_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_5024_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_5025_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_5026_inverse__zero,axiom,
    ( ( inverse_inverse_real @ zero_zero_real )
    = zero_zero_real ) ).

% inverse_zero
thf(fact_5027_inverse__zero,axiom,
    ( ( inverse_inverse_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% inverse_zero
thf(fact_5028_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( ( inverse_inverse_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_5029_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_5030_inverse__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( invers8013647133539491842omplex @ X )
        = one_one_complex )
      = ( X = one_one_complex ) ) ).

% inverse_eq_1_iff
thf(fact_5031_inverse__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( inverse_inverse_real @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% inverse_eq_1_iff
thf(fact_5032_inverse__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( inverse_inverse_rat @ X )
        = one_one_rat )
      = ( X = one_one_rat ) ) ).

% inverse_eq_1_iff
thf(fact_5033_inverse__1,axiom,
    ( ( invers8013647133539491842omplex @ one_one_complex )
    = one_one_complex ) ).

% inverse_1
thf(fact_5034_inverse__1,axiom,
    ( ( inverse_inverse_real @ one_one_real )
    = one_one_real ) ).

% inverse_1
thf(fact_5035_inverse__1,axiom,
    ( ( inverse_inverse_rat @ one_one_rat )
    = one_one_rat ) ).

% inverse_1
thf(fact_5036_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_5037_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_5038_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_5039_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_5040_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_real @ B @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_5041_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_rat @ B @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_5042_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_real @ B @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_5043_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_rat @ B @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_5044_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% inverse_negative_iff_negative
thf(fact_5045_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% inverse_negative_iff_negative
thf(fact_5046_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_5047_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_5048_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_eq_real @ B @ A ) ) ) ) ).

% inverse_le_iff_le
thf(fact_5049_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% inverse_le_iff_le
thf(fact_5050_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_eq_real @ B @ A ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_5051_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_5052_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
        = one_one_complex ) ) ).

% left_inverse
thf(fact_5053_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% left_inverse
thf(fact_5054_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% left_inverse
thf(fact_5055_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
        = one_one_complex ) ) ).

% right_inverse
thf(fact_5056_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
        = one_one_real ) ) ).

% right_inverse
thf(fact_5057_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
        = one_one_rat ) ) ).

% right_inverse
thf(fact_5058_field__class_Ofield__inverse__zero,axiom,
    ( ( inverse_inverse_real @ zero_zero_real )
    = zero_zero_real ) ).

% field_class.field_inverse_zero
thf(fact_5059_field__class_Ofield__inverse__zero,axiom,
    ( ( inverse_inverse_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% field_class.field_inverse_zero
thf(fact_5060_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( ( inverse_inverse_real @ A )
        = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
thf(fact_5061_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
thf(fact_5062_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B ) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_5063_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = ( inverse_inverse_rat @ B ) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_5064_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_5065_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_5066_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ A )
       != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_5067_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ A )
       != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_5068_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_5069_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_5070_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_5071_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_5072_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_5073_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ B @ A ) ) ) ).

% inverse_less_imp_less
thf(fact_5074_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ B @ A ) ) ) ).

% inverse_less_imp_less
thf(fact_5075_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% less_imp_inverse_less
thf(fact_5076_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% less_imp_inverse_less
thf(fact_5077_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ B @ A ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_5078_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ B @ A ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_5079_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_5080_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_5081_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
     => ( ( A != zero_zero_real )
       => ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% inverse_negative_imp_negative
thf(fact_5082_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
     => ( ( A != zero_zero_rat )
       => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% inverse_negative_imp_negative
thf(fact_5083_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
     => ( ( A != zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ A ) ) ) ).

% inverse_positive_imp_positive
thf(fact_5084_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
     => ( ( A != zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).

% inverse_positive_imp_positive
thf(fact_5085_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).

% negative_imp_inverse_negative
thf(fact_5086_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).

% negative_imp_inverse_negative
thf(fact_5087_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).

% positive_imp_inverse_positive
thf(fact_5088_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).

% positive_imp_inverse_positive
thf(fact_5089_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
          = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_5090_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
          = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_5091_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
        = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_5092_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
        = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_5093_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
        = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_5094_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = one_one_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = B ) ) ).

% inverse_unique
thf(fact_5095_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = one_one_real )
     => ( ( inverse_inverse_real @ A )
        = B ) ) ).

% inverse_unique
thf(fact_5096_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = one_one_rat )
     => ( ( inverse_inverse_rat @ A )
        = B ) ) ).

% inverse_unique
thf(fact_5097_inverse__eq__divide,axiom,
    ( invers8013647133539491842omplex
    = ( divide1717551699836669952omplex @ one_one_complex ) ) ).

% inverse_eq_divide
thf(fact_5098_inverse__eq__divide,axiom,
    ( inverse_inverse_real
    = ( divide_divide_real @ one_one_real ) ) ).

% inverse_eq_divide
thf(fact_5099_inverse__eq__divide,axiom,
    ( inverse_inverse_rat
    = ( divide_divide_rat @ one_one_rat ) ) ).

% inverse_eq_divide
thf(fact_5100_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
        = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).

% nonzero_abs_inverse
thf(fact_5101_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
        = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).

% nonzero_abs_inverse
thf(fact_5102_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_5103_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_5104_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X3: real,Y2: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).

% divide_real_def
thf(fact_5105_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_5106_inverse__le__imp__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ B @ A ) ) ) ).

% inverse_le_imp_le
thf(fact_5107_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ B @ A ) ) ) ).

% inverse_le_imp_le
thf(fact_5108_le__imp__inverse__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% le_imp_inverse_le
thf(fact_5109_le__imp__inverse__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% le_imp_inverse_le
thf(fact_5110_inverse__le__imp__le__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ B @ A ) ) ) ).

% inverse_le_imp_le_neg
thf(fact_5111_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ B @ A ) ) ) ).

% inverse_le_imp_le_neg
thf(fact_5112_le__imp__inverse__le__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% le_imp_inverse_le_neg
thf(fact_5113_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% le_imp_inverse_le_neg
thf(fact_5114_inverse__le__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
      = ( ( ord_less_eq_real @ X @ zero_zero_real )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% inverse_le_1_iff
thf(fact_5115_inverse__le__1__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% inverse_le_1_iff
thf(fact_5116_one__less__inverse,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% one_less_inverse
thf(fact_5117_one__less__inverse,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% one_less_inverse
thf(fact_5118_one__less__inverse__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
      = ( ( ord_less_real @ zero_zero_real @ X )
        & ( ord_less_real @ X @ one_one_real ) ) ) ).

% one_less_inverse_iff
thf(fact_5119_one__less__inverse__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X )
        & ( ord_less_rat @ X @ one_one_rat ) ) ) ).

% one_less_inverse_iff
thf(fact_5120_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% inverse_add
thf(fact_5121_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% inverse_add
thf(fact_5122_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_5123_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_5124_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
        = one_one_complex ) ) ).

% field_class.field_inverse
thf(fact_5125_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% field_class.field_inverse
thf(fact_5126_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% field_class.field_inverse
thf(fact_5127_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_5128_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_5129_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_5130_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ A )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_5131_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ A )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_5132_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).

% inverse_powr
thf(fact_5133_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).

% vebt_member.simps(3)
thf(fact_5134_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_5135_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_5136_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_5137_inverse__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ B @ A ) )
        & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
         => ( ord_less_eq_real @ A @ B ) ) ) ) ).

% inverse_le_iff
thf(fact_5138_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ B @ A ) )
        & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
         => ( ord_less_eq_rat @ A @ B ) ) ) ) ).

% inverse_le_iff
thf(fact_5139_inverse__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ B @ A ) )
        & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
         => ( ord_less_real @ A @ B ) ) ) ) ).

% inverse_less_iff
thf(fact_5140_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ B @ A ) )
        & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
         => ( ord_less_rat @ A @ B ) ) ) ) ).

% inverse_less_iff
thf(fact_5141_one__le__inverse,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% one_le_inverse
thf(fact_5142_one__le__inverse,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% one_le_inverse
thf(fact_5143_inverse__less__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
      = ( ( ord_less_eq_real @ X @ zero_zero_real )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% inverse_less_1_iff
thf(fact_5144_inverse__less__1__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% inverse_less_1_iff
thf(fact_5145_one__le__inverse__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
      = ( ( ord_less_real @ zero_zero_real @ X )
        & ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% one_le_inverse_iff
thf(fact_5146_one__le__inverse__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X )
        & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).

% one_le_inverse_iff
thf(fact_5147_inverse__diff__inverse,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
          = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_5148_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_5149_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_5150_reals__Archimedean,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N2: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ X ) ) ).

% reals_Archimedean
thf(fact_5151_reals__Archimedean,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N2: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ X ) ) ).

% reals_Archimedean
thf(fact_5152_forall__pos__mono__1,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D6: real,E: real] :
          ( ( ord_less_real @ D6 @ E )
         => ( ( P @ D6 )
           => ( P @ E ) ) )
     => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_5153_real__arch__inverse,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
      = ( ? [N4: nat] :
            ( ( N4 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E2 ) ) ) ) ).

% real_arch_inverse
thf(fact_5154_forall__pos__mono,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D6: real,E: real] :
          ( ( ord_less_real @ D6 @ E )
         => ( ( P @ D6 )
           => ( P @ E ) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono
thf(fact_5155_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).

% vebt_member.simps(4)
thf(fact_5156_ex__inverse__of__nat__less,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X ) ) ) ).

% ex_inverse_of_nat_less
thf(fact_5157_ex__inverse__of__nat__less,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ X ) ) ) ).

% ex_inverse_of_nat_less
thf(fact_5158_power__diff__conv__inverse,axiom,
    ! [X: complex,M2: nat,N: nat] :
      ( ( X != zero_zero_complex )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M2 ) )
          = ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M2 ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_5159_power__diff__conv__inverse,axiom,
    ! [X: real,M2: nat,N: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M2 ) )
          = ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M2 ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_5160_power__diff__conv__inverse,axiom,
    ! [X: rat,M2: nat,N: nat] :
      ( ( X != zero_zero_rat )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M2 ) )
          = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M2 ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_5161_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_5162_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_5163_vebt__pred_Osimps_I3_J,axiom,
    ! [B: $o,A: $o,Va2: nat] :
      ( ( B
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_5164_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_5165_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_5166_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Ma @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_5167_option_Osize__gen_I2_J,axiom,
    ! [X: nat > nat,X23: nat] :
      ( ( size_option_nat @ X @ ( some_nat @ X23 ) )
      = ( plus_plus_nat @ ( X @ X23 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_5168_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X23: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X23 ) )
      = ( plus_plus_nat @ ( X @ X23 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_5169_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X23: num] :
      ( ( size_option_num @ X @ ( some_num @ X23 ) )
      = ( plus_plus_nat @ ( X @ X23 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_5170_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
            & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_5171_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [J3: nat] :
        ? [M8: nat] :
        ! [M3: nat] :
          ( ( ord_less_eq_nat @ M8 @ M3 )
         => ! [N4: nat] :
              ( ( ord_less_eq_nat @ M8 @ N4 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_5172_option_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( size_option_nat @ X @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_5173_option_Osize__gen_I1_J,axiom,
    ! [X: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_5174_option_Osize__gen_I1_J,axiom,
    ! [X: num > nat] :
      ( ( size_option_num @ X @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_5175_of__int__floor__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
        = X )
      = ( ? [N4: int] :
            ( X
            = ( ring_1_of_int_real @ N4 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_5176_of__int__floor__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
        = X )
      = ( ? [N4: int] :
            ( X
            = ( ring_1_of_int_rat @ N4 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_5177_floor__zero,axiom,
    ( ( archim6058952711729229775r_real @ zero_zero_real )
    = zero_zero_int ) ).

% floor_zero
thf(fact_5178_floor__zero,axiom,
    ( ( archim3151403230148437115or_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% floor_zero
thf(fact_5179_floor__one,axiom,
    ( ( archim6058952711729229775r_real @ one_one_real )
    = one_one_int ) ).

% floor_one
thf(fact_5180_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_5181_zero__le__floor,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_floor
thf(fact_5182_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_5183_floor__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% floor_less_zero
thf(fact_5184_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_5185_zero__less__floor,axiom,
    ! [X: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ one_one_real @ X ) ) ).

% zero_less_floor
thf(fact_5186_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_5187_floor__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% floor_le_zero
thf(fact_5188_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_5189_one__le__floor,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ one_one_real @ X ) ) ).

% one_le_floor
thf(fact_5190_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_5191_floor__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% floor_less_one
thf(fact_5192_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_5193_floor__diff__one,axiom,
    ! [X: real] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_5194_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_5195_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A5: $o,B5: $o] :
          ( X
         != ( vEBT_Leaf @ A5 @ B5 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_mint.cases
thf(fact_5196_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ S ) )
     => ( ord_le3000389064537975527at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_5197_subrelI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ! [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y3 ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y3 ) @ S ) )
     => ( ord_le5997549366648089703at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_5198_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ S ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_5199_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_5200_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R2 )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S ) )
     => ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).

% subrelI
thf(fact_5201_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).

% floor_mono
thf(fact_5202_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).

% floor_mono
thf(fact_5203_of__int__floor__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_5204_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_5205_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_5206_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_5207_floor__le__ceiling,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).

% floor_le_ceiling
thf(fact_5208_floor__le__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).

% floor_le_ceiling
thf(fact_5209_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
thf(fact_5210_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_5211_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_5212_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_5213_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_5214_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_5215_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).

% vebt_member.simps(2)
thf(fact_5216_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_5217_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_5218_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_5219_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).

% floor_less_iff
thf(fact_5220_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% floor_less_iff
thf(fact_5221_le__floor__add,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_5222_le__floor__add,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_5223_floor__power,axiom,
    ! [X: real,N: nat] :
      ( ( X
        = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
     => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N ) )
        = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N ) ) ) ).

% floor_power
thf(fact_5224_floor__power,axiom,
    ! [X: rat,N: nat] :
      ( ( X
        = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
     => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N ) )
        = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N ) ) ) ).

% floor_power
thf(fact_5225_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( A5
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A5
                 => ( ( B5
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B5
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_5226_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( B5
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B5
                 => ( ( A5
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A5
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_5227_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_5228_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_5229_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_5230_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( X
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_5231_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv2: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y )
         => ( ( ? [Uu2: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_5232_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va2 )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_5233_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) @ Vb )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_5234_of__nat__floor,axiom,
    ! [R2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).

% of_nat_floor
thf(fact_5235_of__nat__floor,axiom,
    ! [R2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).

% of_nat_floor
thf(fact_5236_one__add__floor,axiom,
    ! [X: real] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% one_add_floor
thf(fact_5237_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_5238_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).

% le_mult_nat_floor
thf(fact_5239_le__mult__nat__floor,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% le_mult_nat_floor
thf(fact_5240_nat__floor__neg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_5241_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_5242_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
     => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_5243_ceiling__diff__floor__le__1,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_5244_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_5245_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_5246_floor__eq,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq
thf(fact_5247_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_5248_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_5249_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_5250_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_5251_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_5252_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
     => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_5253_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( ( archim6058952711729229775r_real @ X )
        = A )
      = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).

% floor_eq_iff
thf(fact_5254_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_5255_floor__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I4 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) ) )
           => ( P @ I4 ) ) ) ) ).

% floor_split
thf(fact_5256_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_5257_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_5258_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_5259_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).

% less_floor_iff
thf(fact_5260_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% less_floor_iff
thf(fact_5261_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_5262_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_5263_floor__correct,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_5264_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_5265_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_5266_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq2
thf(fact_5267_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).

% floor_divide_real_eq_div
thf(fact_5268_floor__divide__lower,axiom,
    ! [Q4: real,P6: real] :
      ( ( ord_less_real @ zero_zero_real @ Q4 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P6 @ Q4 ) ) ) @ Q4 ) @ P6 ) ) ).

% floor_divide_lower
thf(fact_5269_floor__divide__lower,axiom,
    ! [Q4: rat,P6: 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 @ P6 @ Q4 ) ) ) @ Q4 ) @ P6 ) ) ).

% floor_divide_lower
thf(fact_5270_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5271_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5272_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5273_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_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5274_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_5275_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_5276_floor__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_5277_floor__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_5278_floor__divide__upper,axiom,
    ! [Q4: real,P6: real] :
      ( ( ord_less_real @ zero_zero_real @ Q4 )
     => ( ord_less_real @ P6 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P6 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) ) ) ).

% floor_divide_upper
thf(fact_5279_floor__divide__upper,axiom,
    ! [Q4: rat,P6: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q4 )
     => ( ord_less_rat @ P6 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P6 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) ) ) ).

% floor_divide_upper
thf(fact_5280_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_5281_divides__aux__eq,axiom,
    ! [Q4: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_5282_divides__aux__eq,axiom,
    ! [Q4: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_5283_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_5284_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K3 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_5285_gbinomial__pochhammer_H,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_5286_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_5287_gbinomial__pochhammer_H,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_5288_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A4 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_5289_gbinomial__pochhammer,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A4 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_5290_gbinomial__pochhammer,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A4 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_5291_sinh__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sinh_real @ X )
        = zero_zero_real )
      = ( member_real @ ( exp_real @ X ) @ ( insert_real @ one_one_real @ ( insert_real @ ( uminus_uminus_real @ one_one_real ) @ bot_bot_set_real ) ) ) ) ).

% sinh_zero_iff
thf(fact_5292_sinh__zero__iff,axiom,
    ! [X: complex] :
      ( ( ( sinh_complex @ X )
        = zero_zero_complex )
      = ( member_complex @ ( exp_complex @ X ) @ ( insert_complex @ one_one_complex @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ bot_bot_set_complex ) ) ) ) ).

% sinh_zero_iff
thf(fact_5293_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% sinh_real_le_iff
thf(fact_5294_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% sinh_real_nonneg_iff
thf(fact_5295_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_5296_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% List.finite_set
thf(fact_5297_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_5298_List_Ofinite__set,axiom,
    ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).

% List.finite_set
thf(fact_5299_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).

% List.finite_set
thf(fact_5300_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).

% List.finite_set
thf(fact_5301_List_Ofinite__set,axiom,
    ! [Xs: list_Extended_enat] : ( finite4001608067531595151d_enat @ ( set_Extended_enat2 @ Xs ) ) ).

% List.finite_set
thf(fact_5302_sinh__0,axiom,
    ( ( sinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% sinh_0
thf(fact_5303_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_5304_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_5305_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_5306_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_5307_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_5308_fact__1,axiom,
    ( ( semiri5044797733671781792omplex @ one_one_nat )
    = one_one_complex ) ).

% fact_1
thf(fact_5309_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_5310_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_5311_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_5312_fact__1,axiom,
    ( ( semiri2265585572941072030t_real @ one_one_nat )
    = one_one_real ) ).

% fact_1
thf(fact_5313_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_5314_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_5315_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_5316_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_5317_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_5318_fact__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono_nat
thf(fact_5319_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_5320_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ N )
     != zero_zero_rat ) ).

% fact_nonzero
thf(fact_5321_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ N )
     != zero_zero_int ) ).

% fact_nonzero
thf(fact_5322_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ N )
     != zero_zero_nat ) ).

% fact_nonzero
thf(fact_5323_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ N )
     != zero_zero_real ) ).

% fact_nonzero
thf(fact_5324_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5325_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5326_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5327_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5328_finite__list,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ? [Xs3: list_P6011104703257516679at_nat] :
          ( ( set_Pr5648618587558075414at_nat @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5329_finite__list,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ? [Xs3: list_Extended_enat] :
          ( ( set_Extended_enat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5330_subset__code_I1_J,axiom,
    ! [Xs: list_o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B2 )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
           => ( member_o @ X3 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_5331_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
           => ( member_set_nat @ X3 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_5332_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat_rat,B2: set_set_nat_rat] :
      ( ( ord_le4375437777232675859at_rat @ ( set_set_nat_rat2 @ Xs ) @ B2 )
      = ( ! [X3: set_nat_rat] :
            ( ( member_set_nat_rat @ X3 @ ( set_set_nat_rat2 @ Xs ) )
           => ( member_set_nat_rat @ X3 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_5333_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
      = ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( member_VEBT_VEBT @ X3 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_5334_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X3 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_5335_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
      = ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
           => ( member_int @ X3 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_5336_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F4: nat > nat > nat,A5: nat,B5: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F4 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_5337_fact__less__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_5338_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_zero
thf(fact_5339_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_zero
thf(fact_5340_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_zero
thf(fact_5341_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_zero
thf(fact_5342_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_gt_zero
thf(fact_5343_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_gt_zero
thf(fact_5344_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_gt_zero
thf(fact_5345_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_gt_zero
thf(fact_5346_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_5347_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_5348_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_5349_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_5350_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_1
thf(fact_5351_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_1
thf(fact_5352_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_1
thf(fact_5353_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_1
thf(fact_5354_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_mono
thf(fact_5355_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M2 ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_mono
thf(fact_5356_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono
thf(fact_5357_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M2 ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_mono
thf(fact_5358_pochhammer__fact,axiom,
    ( semiri5044797733671781792omplex
    = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).

% pochhammer_fact
thf(fact_5359_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_5360_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_5361_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_5362_pochhammer__fact,axiom,
    ( semiri2265585572941072030t_real
    = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).

% pochhammer_fact
thf(fact_5363_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D6: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D6 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_5364_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_5365_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_5366_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M2 ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).

% fact_less_mono
thf(fact_5367_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_5368_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M2 ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).

% fact_less_mono
thf(fact_5369_length__pos__if__in__set,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5370_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5371_length__pos__if__in__set,axiom,
    ! [X: set_nat_rat,Xs: list_set_nat_rat] :
      ( ( member_set_nat_rat @ X @ ( set_set_nat_rat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3959913991096427681at_rat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5372_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5373_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5374_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5375_card__length,axiom,
    ! [Xs: list_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( set_complex2 @ Xs ) ) @ ( size_s3451745648224563538omplex @ Xs ) ) ).

% card_length
thf(fact_5376_card__length,axiom,
    ! [Xs: list_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( set_list_nat2 @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).

% card_length
thf(fact_5377_card__length,axiom,
    ! [Xs: list_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ).

% card_length
thf(fact_5378_card__length,axiom,
    ! [Xs: list_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( set_int2 @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ).

% card_length
thf(fact_5379_card__length,axiom,
    ! [Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% card_length
thf(fact_5380_card__length,axiom,
    ! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% card_length
thf(fact_5381_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_5382_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_5383_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_5384_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_5385_fact__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ ( suc @ M2 ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ N ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_5386_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_5387_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X4: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X4 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ X4 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_5388_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M3 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5389_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5390_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5391_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5392_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5393_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_5394_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_5395_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_5396_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri2265585572941072030t_real @ N )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_5397_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% pochhammer_same
thf(fact_5398_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_5399_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_5400_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% pochhammer_same
thf(fact_5401_vebt__delete_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ zero_zero_nat ) )
     => ( ! [A5: $o,B5: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A5: $o,B5: $o,N2: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) )
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X4 ) )
             => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X4: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X4 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).

% vebt_delete.cases
thf(fact_5402_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ X4 ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X4 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_5403_vebt__pred_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A5: $o,Uw2: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A5: $o,B5: $o,Va: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).

% vebt_pred.cases
thf(fact_5404_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B5: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw2: $o,N2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_5405_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X4: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X4 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_5406_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X4: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X4 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) @ X4 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ X4 ) )
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_5407_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( B5
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B5
                   => ( ( A5
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A5
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_5408_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( A5
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A5
                   => ( ( B5
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B5
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_5409_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q4 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_5410_set__removeAll,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( remove3673390508374433037at_nat @ X @ Xs ) )
      = ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).

% set_removeAll
thf(fact_5411_set__removeAll,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( removeAll_VEBT_VEBT @ X @ Xs ) )
      = ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_removeAll
thf(fact_5412_set__removeAll,axiom,
    ! [X: real,Xs: list_real] :
      ( ( set_real2 @ ( removeAll_real @ X @ Xs ) )
      = ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% set_removeAll
thf(fact_5413_set__removeAll,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( set_o2 @ ( removeAll_o @ X @ Xs ) )
      = ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% set_removeAll
thf(fact_5414_set__removeAll,axiom,
    ! [X: int,Xs: list_int] :
      ( ( set_int2 @ ( removeAll_int @ X @ Xs ) )
      = ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% set_removeAll
thf(fact_5415_set__removeAll,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( set_nat2 @ ( removeAll_nat @ X @ Xs ) )
      = ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% set_removeAll
thf(fact_5416_inthall,axiom,
    ! [Xs: list_o,P: $o > $o,N: nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_5417_inthall,axiom,
    ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
       => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_5418_inthall,axiom,
    ! [Xs: list_set_nat_rat,P: set_nat_rat > $o,N: nat] :
      ( ! [X4: set_nat_rat] :
          ( ( member_set_nat_rat @ X4 @ ( set_set_nat_rat2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N @ ( size_s3959913991096427681at_rat @ Xs ) )
       => ( P @ ( nth_set_nat_rat @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_5419_inthall,axiom,
    ! [Xs: list_int,P: int > $o,N: nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_5420_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_5421_inthall,axiom,
    ! [Xs: list_nat,P: nat > $o,N: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_5422_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I2 )
              = ( nth_int @ Ys2 @ I2 ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_5423_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( ( nth_VEBT_VEBT @ Xs @ I2 )
              = ( nth_VEBT_VEBT @ Ys2 @ I2 ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_5424_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys2 @ I2 ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_5425_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X8: int] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs2: list_int] :
            ( ( ( size_size_list_int @ Xs2 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_5426_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X8: vEBT_VEBT] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs2: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_5427_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X8: nat] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_5428_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_int,Z4: list_int] : ( Y5 = Z4 ) )
    = ( ^ [Xs2: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs2 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
             => ( ( nth_int @ Xs2 @ I4 )
                = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_5429_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : ( Y5 = Z4 ) )
    = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
                = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_5430_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z4: list_nat] : ( Y5 = Z4 ) )
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I4 )
                = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_5431_length__removeAll__less__eq,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_5432_length__removeAll__less__eq,axiom,
    ! [X: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_5433_nth__mem,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).

% nth_mem
thf(fact_5434_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_5435_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat_rat] :
      ( ( ord_less_nat @ N @ ( size_s3959913991096427681at_rat @ Xs ) )
     => ( member_set_nat_rat @ ( nth_set_nat_rat @ Xs @ N ) @ ( set_set_nat_rat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_5436_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).

% nth_mem
thf(fact_5437_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% nth_mem
thf(fact_5438_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_5439_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_5440_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_5441_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_5442_in__set__conv__nth,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
            & ( ( nth_o @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_5443_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
            & ( ( nth_set_nat @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_5444_in__set__conv__nth,axiom,
    ! [X: set_nat_rat,Xs: list_set_nat_rat] :
      ( ( member_set_nat_rat @ X @ ( set_set_nat_rat2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3959913991096427681at_rat @ Xs ) )
            & ( ( nth_set_nat_rat @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_5445_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_5446_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_5447_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_5448_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: $o > $o,X: $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
         => ( P @ ( nth_o @ Xs @ I2 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_5449_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
         => ( P @ ( nth_set_nat @ Xs @ I2 ) ) )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_5450_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat_rat,P: set_nat_rat > $o,X: set_nat_rat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3959913991096427681at_rat @ Xs ) )
         => ( P @ ( nth_set_nat_rat @ Xs @ I2 ) ) )
     => ( ( member_set_nat_rat @ X @ ( set_set_nat_rat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_5451_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: int > $o,X: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I2 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_5452_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_5453_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I2 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_5454_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
           => ( P @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_5455_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_5456_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_5457_length__removeAll__less,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_o @ ( removeAll_o @ X @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_5458_length__removeAll__less,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
     => ( ord_less_nat @ ( size_s3254054031482475050et_nat @ ( removeAll_set_nat @ X @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_5459_length__removeAll__less,axiom,
    ! [X: set_nat_rat,Xs: list_set_nat_rat] :
      ( ( member_set_nat_rat @ X @ ( set_set_nat_rat2 @ Xs ) )
     => ( ord_less_nat @ ( size_s3959913991096427681at_rat @ ( remove939820145577552881at_rat @ X @ Xs ) ) @ ( size_s3959913991096427681at_rat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_5460_length__removeAll__less,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_int @ ( removeAll_int @ X @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_5461_length__removeAll__less,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_5462_length__removeAll__less,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_5463_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_5464_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_5465_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ! [Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_5466_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_5467_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_int,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_int @ Xs ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M2 )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M2 ) @ ( nth_int @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_5468_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_VEBT_VEBT,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) @ M2 )
        = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M2 ) @ ( nth_VEBT_VEBT @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_5469_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M2 )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M2 ) @ ( nth_nat @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_5470_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_5471_length__mul__elem,axiom,
    ! [Xs: list_list_VEBT_VEBT,N: nat] :
      ( ! [X4: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X4 @ ( set_list_VEBT_VEBT2 @ Xs ) )
         => ( ( size_s6755466524823107622T_VEBT @ X4 )
            = N ) )
     => ( ( size_s6755466524823107622T_VEBT @ ( concat_VEBT_VEBT @ Xs ) )
        = ( times_times_nat @ ( size_s8217280938318005548T_VEBT @ Xs ) @ N ) ) ) ).

% length_mul_elem
thf(fact_5472_length__mul__elem,axiom,
    ! [Xs: list_list_nat,N: nat] :
      ( ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
         => ( ( size_size_list_nat @ X4 )
            = N ) )
     => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
        = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ) ).

% length_mul_elem
thf(fact_5473_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_5474_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_5475_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: $o,B: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_5476_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
       => ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys2 ) @ I )
          = ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5477_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
       => ( ( nth_Pr3474266648193625910T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys2 ) @ I )
          = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5478_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
       => ( ( nth_Pr8617346907841251940nt_nat @ ( zip_int_nat @ Xs @ Ys2 ) @ I )
          = ( product_Pair_int_nat @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5479_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
       => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys2 ) @ I )
          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5480_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
       => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys2 ) @ I )
          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5481_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
       => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys2 ) @ I )
          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5482_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys2: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
       => ( ( nth_Pr3440142176431000676at_int @ ( zip_nat_int @ Xs @ Ys2 ) @ I )
          = ( product_Pair_nat_int @ ( nth_nat @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5483_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
       => ( ( nth_Pr744662078594809490T_VEBT @ ( zip_nat_VEBT_VEBT @ Xs @ Ys2 ) @ I )
          = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5484_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) @ I )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5485_nth__zip,axiom,
    ! [I: nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys2 ) )
       => ( ( nth_Pr6744343527793145070at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 ) @ I )
          = ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I ) @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5486_find__Some__iff,axiom,
    ! [P: int > $o,Xs: list_int,X: int] :
      ( ( ( find_int @ P @ Xs )
        = ( some_int @ X ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
            & ( P @ ( nth_int @ Xs @ I4 ) )
            & ( X
              = ( nth_int @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5487_find__Some__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
        = ( some_P7363390416028606310at_nat @ X ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
            & ( X
              = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5488_find__Some__iff,axiom,
    ! [P: num > $o,Xs: list_num,X: num] :
      ( ( ( find_num @ P @ Xs )
        = ( some_num @ X ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
            & ( P @ ( nth_num @ Xs @ I4 ) )
            & ( X
              = ( nth_num @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5489_find__Some__iff,axiom,
    ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( find_VEBT_VEBT @ P @ Xs )
        = ( some_VEBT_VEBT @ X ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
            & ( X
              = ( nth_VEBT_VEBT @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5490_find__Some__iff,axiom,
    ! [P: nat > $o,Xs: list_nat,X: nat] :
      ( ( ( find_nat @ P @ Xs )
        = ( some_nat @ X ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I4 ) )
            & ( X
              = ( nth_nat @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5491_find__Some__iff2,axiom,
    ! [X: int,P: int > $o,Xs: list_int] :
      ( ( ( some_int @ X )
        = ( find_int @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
            & ( P @ ( nth_int @ Xs @ I4 ) )
            & ( X
              = ( nth_int @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5492_find__Some__iff2,axiom,
    ! [X: product_prod_nat_nat,P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X )
        = ( find_P8199882355184865565at_nat @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
            & ( X
              = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5493_find__Some__iff2,axiom,
    ! [X: num,P: num > $o,Xs: list_num] :
      ( ( ( some_num @ X )
        = ( find_num @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
            & ( P @ ( nth_num @ Xs @ I4 ) )
            & ( X
              = ( nth_num @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5494_find__Some__iff2,axiom,
    ! [X: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( ( some_VEBT_VEBT @ X )
        = ( find_VEBT_VEBT @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
            & ( X
              = ( nth_VEBT_VEBT @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5495_find__Some__iff2,axiom,
    ! [X: nat,P: nat > $o,Xs: list_nat] :
      ( ( ( some_nat @ X )
        = ( find_nat @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I4 ) )
            & ( X
              = ( nth_nat @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5496_nth__Cons__pos,axiom,
    ! [N: nat,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
        = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_5497_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_5498_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_5499_rotate1__length01,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat )
     => ( ( rotate1_VEBT_VEBT @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_5500_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_5501_remove__def,axiom,
    ( remove6466555014256735590at_nat
    = ( ^ [X3: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] : ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% remove_def
thf(fact_5502_remove__def,axiom,
    ( remove_real
    = ( ^ [X3: real,A6: set_real] : ( minus_minus_set_real @ A6 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).

% remove_def
thf(fact_5503_remove__def,axiom,
    ( remove_o
    = ( ^ [X3: $o,A6: set_o] : ( minus_minus_set_o @ A6 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).

% remove_def
thf(fact_5504_remove__def,axiom,
    ( remove_int
    = ( ^ [X3: int,A6: set_int] : ( minus_minus_set_int @ A6 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).

% remove_def
thf(fact_5505_remove__def,axiom,
    ( remove_nat
    = ( ^ [X3: nat,A6: set_nat] : ( minus_minus_set_nat @ A6 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% remove_def
thf(fact_5506_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_5507_member__remove,axiom,
    ! [X: $o,Y: $o,A2: set_o] :
      ( ( member_o @ X @ ( remove_o @ Y @ A2 ) )
      = ( ( member_o @ X @ A2 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_5508_member__remove,axiom,
    ! [X: set_nat,Y: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ X @ ( remove_set_nat @ Y @ A2 ) )
      = ( ( member_set_nat @ X @ A2 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_5509_member__remove,axiom,
    ! [X: set_nat_rat,Y: set_nat_rat,A2: set_set_nat_rat] :
      ( ( member_set_nat_rat @ X @ ( remove_set_nat_rat @ Y @ A2 ) )
      = ( ( member_set_nat_rat @ X @ A2 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_5510_member__remove,axiom,
    ! [X: nat,Y: nat,A2: set_nat] :
      ( ( member_nat @ X @ ( remove_nat @ Y @ A2 ) )
      = ( ( member_nat @ X @ A2 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_5511_member__remove,axiom,
    ! [X: int,Y: int,A2: set_int] :
      ( ( member_int @ X @ ( remove_int @ Y @ A2 ) )
      = ( ( member_int @ X @ A2 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_5512_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_5513_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_5514_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_5515_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_5516_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_5517_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_5518_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_5519_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_5520_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_5521_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_5522_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_5523_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_5524_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_5525_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_5526_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_5527_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_5528_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_5529_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_5530_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_5531_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_5532_nth__Cons__0,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_5533_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_5534_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_5535_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_5536_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_5537_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_5538_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_5539_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_5540_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_5541_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_5542_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_5543_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_5544_zmod__le__nonneg__dividend,axiom,
    ! [M2: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M2 )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M2 @ K ) @ M2 ) ) ).

% zmod_le_nonneg_dividend
thf(fact_5545_set__subset__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5546_set__subset__Cons,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5547_set__subset__Cons,axiom,
    ! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5548_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_5549_impossible__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
     => ( Xs
       != ( cons_VEBT_VEBT @ X @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_5550_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_5551_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_5552_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_5553_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_5554_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_5555_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_5556_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_5557_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_5558_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_5559_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_5560_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_5561_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_5562_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_5563_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_5564_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_5565_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_5566_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_5567_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_5568_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_5569_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_5570_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_5571_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_5572_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_5573_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_5574_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_5575_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_5576_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_5577_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M2 ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_5578_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M2 ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_5579_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_5580_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_5581_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_5582_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_5583_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_5584_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
      = ( ? [X3: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5585_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) )
      = ( ? [X3: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
            ( ( Xs
              = ( cons_VEBT_VEBT @ X3 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5586_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X3: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5587_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_5588_list_Osize_I4_J,axiom,
    ! [X21: int,X22: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_int @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5589_list_Osize_I4_J,axiom,
    ! [X21: vEBT_VEBT,X22: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5590_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5591_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_5592_nth__Cons_H,axiom,
    ! [N: nat,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
          = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_5593_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_5594_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_5595_real__of__int__div__aux,axiom,
    ! [X: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_5596_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_5597_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_5598_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_5599_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_5600_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_5601_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_5602_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_5603_nth__equal__first__eq,axiom,
    ! [X: $o,Xs: list_o,N: nat] :
      ( ~ ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( ( ( nth_o @ ( cons_o @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5604_nth__equal__first__eq,axiom,
    ! [X: set_nat,Xs: list_set_nat,N: nat] :
      ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
       => ( ( ( nth_set_nat @ ( cons_set_nat @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5605_nth__equal__first__eq,axiom,
    ! [X: set_nat_rat,Xs: list_set_nat_rat,N: nat] :
      ( ~ ( member_set_nat_rat @ X @ ( set_set_nat_rat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3959913991096427681at_rat @ Xs ) )
       => ( ( ( nth_set_nat_rat @ ( cons_set_nat_rat @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5606_nth__equal__first__eq,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5607_nth__equal__first__eq,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5608_nth__equal__first__eq,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5609_nth__non__equal__first__eq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5610_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5611_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5612_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B2 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B2 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_5613_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_5614_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B2 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B2 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_5615_norm__power__diff,axiom,
    ! [Z: real,W2: real,M2: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W2 ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M2 ) @ ( power_power_real @ W2 @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W2 ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_5616_norm__power__diff,axiom,
    ! [Z: complex,W2: complex,M2: nat] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M2 ) @ ( power_power_complex @ W2 @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W2 ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_5617_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5618_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_5619_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5620_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_5621_Gcd__fin__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( ( semiri4258706085729940814in_nat @ A2 )
        = zero_zero_nat )
      = ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
        & ( finite_finite_nat @ A2 ) ) ) ).

% Gcd_fin_0_iff
thf(fact_5622_Gcd__fin__0__iff,axiom,
    ! [A2: set_int] :
      ( ( ( semiri4256215615220890538in_int @ A2 )
        = zero_zero_int )
      = ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
        & ( finite_finite_int @ A2 ) ) ) ).

% Gcd_fin_0_iff
thf(fact_5623_count__list_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      ( ( ( X = Y )
       => ( ( count_list_int @ ( cons_int @ X @ Xs ) @ Y )
          = ( plus_plus_nat @ ( count_list_int @ Xs @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_list_int @ ( cons_int @ X @ Xs ) @ Y )
          = ( count_list_int @ Xs @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_5624_count__list_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      ( ( ( X = Y )
       => ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
          = ( plus_plus_nat @ ( count_list_nat @ Xs @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
          = ( count_list_nat @ Xs @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_5625_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A2 )
                 => ( X2
                   != ( minus_minus_int @ Xb @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X2 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_5626_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_real @ M2 )
        = ( numeral_numeral_real @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_5627_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_nat @ M2 )
        = ( numeral_numeral_nat @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_5628_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_int @ M2 )
        = ( numeral_numeral_int @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_5629_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numera1916890842035813515d_enat @ M2 )
        = ( numera1916890842035813515d_enat @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_5630_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numera6620942414471956472nteger @ M2 )
        = ( numera6620942414471956472nteger @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_5631_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_5632_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_5633_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_5634_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_5635_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_5636_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_5637_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_5638_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_5639_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_5640_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_5641_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_5642_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_5643_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Z ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_5644_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Z ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_5645_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W2 ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_5646_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_5647_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z ) )
      = ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_5648_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ Z ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_5649_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5650_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5651_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5652_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5653_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5654_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5655_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_5656_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_5657_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_5658_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_5659_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_5660_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ W2 ) @ Z ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_5661_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_5662_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_5663_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_5664_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_5665_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_5666_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_5667_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_5668_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_5669_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_5670_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_5671_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_5672_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5673_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5674_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5675_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5676_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5677_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_5678_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri4216267220026989637d_enat @ ( numeral_numeral_nat @ N ) )
      = ( numera1916890842035813515d_enat @ N ) ) ).

% of_nat_numeral
thf(fact_5679_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri4939895301339042750nteger @ ( numeral_numeral_nat @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% of_nat_numeral
thf(fact_5680_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_5681_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_5682_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% of_nat_numeral
thf(fact_5683_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L2: set_int,H2: set_int] :
      ( ( ( set_or370866239135849197et_int @ L @ H )
        = ( set_or370866239135849197et_int @ L2 @ H2 ) )
      = ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ ( ord_less_eq_set_int @ L @ H )
          & ~ ( ord_less_eq_set_int @ L2 @ H2 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5684_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L2: rat,H2: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H )
        = ( set_or633870826150836451st_rat @ L2 @ H2 ) )
      = ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H )
          & ~ ( ord_less_eq_rat @ L2 @ H2 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5685_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L2: num,H2: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H )
        = ( set_or7049704709247886629st_num @ L2 @ H2 ) )
      = ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ ( ord_less_eq_num @ L @ H )
          & ~ ( ord_less_eq_num @ L2 @ H2 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5686_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H )
        = ( set_or1266510415728281911st_int @ L2 @ H2 ) )
      = ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ ( ord_less_eq_int @ L @ H )
          & ~ ( ord_less_eq_int @ L2 @ H2 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5687_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H )
        = ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
      = ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H )
          & ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5688_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L2: real,H2: real] :
      ( ( ( set_or1222579329274155063t_real @ L @ H )
        = ( set_or1222579329274155063t_real @ L2 @ H2 ) )
      = ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ ( ord_less_eq_real @ L @ H )
          & ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5689_atLeastAtMost__iff,axiom,
    ! [I: $o,L: $o,U: $o] :
      ( ( member_o @ I @ ( set_or8904488021354931149Most_o @ L @ U ) )
      = ( ( ord_less_eq_o @ L @ I )
        & ( ord_less_eq_o @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5690_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5691_atLeastAtMost__iff,axiom,
    ! [I: set_nat_rat,L: set_nat_rat,U: set_nat_rat] :
      ( ( member_set_nat_rat @ I @ ( set_or5795412311047298440at_rat @ L @ U ) )
      = ( ( ord_le2679597024174929757at_rat @ L @ I )
        & ( ord_le2679597024174929757at_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5692_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
      = ( ( ord_less_eq_set_int @ L @ I )
        & ( ord_less_eq_set_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5693_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5694_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5695_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5696_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5697_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5698_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_5699_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_numeral
thf(fact_5700_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_5701_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_numeral
thf(fact_5702_mod__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ M2 @ N )
        = M2 ) ) ).

% mod_less
thf(fact_5703_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_5704_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5705_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5706_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5707_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5708_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_5709_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_5710_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_5711_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_5712_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V: num] :
      ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( numera1916890842035813515d_enat @ V ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_5713_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_5714_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_5715_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_5716_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_5717_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_5718_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_5719_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_5720_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5721_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5722_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5723_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5724_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_5725_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_5726_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_5727_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_5728_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_5729_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_5730_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_5731_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_5732_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5733_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5734_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5735_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5736_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5737_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5738_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5739_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5740_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5741_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5742_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5743_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5744_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5745_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5746_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5747_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5748_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5749_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5750_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5751_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5752_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5753_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5754_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5755_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5756_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5757_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5758_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5759_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5760_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5761_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5762_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_5763_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_5764_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_neg_numeral
thf(fact_5765_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_5766_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5767_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( set_or370866239135849197et_int @ A @ B )
        = bot_bot_set_set_int )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5768_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_5769_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_5770_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_5771_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_5772_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5773_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o
        = ( set_or8904488021354931149Most_o @ A @ B ) )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5774_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int
        = ( set_or370866239135849197et_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5775_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_5776_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_5777_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_5778_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_5779_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5780_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( ( ord_less_eq_set_int @ C @ A )
          & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5781_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_5782_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_5783_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_5784_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_5785_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5786_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( ( ord_less_o @ B @ A )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o ) ) ).

% atLeastatMost_empty
thf(fact_5787_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_5788_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_5789_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_5790_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_5791_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_5792_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_5793_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_5794_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_5795_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_5796_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( ( real_V7735802525324610683m_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_5797_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( ( real_V1022390504157884413omplex @ X )
        = zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_5798_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_5799_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_5800_atLeastAtMost__singleton,axiom,
    ! [A: $o] :
      ( ( set_or8904488021354931149Most_o @ A @ A )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% atLeastAtMost_singleton
thf(fact_5801_atLeastAtMost__singleton,axiom,
    ! [A: int] :
      ( ( set_or1266510415728281911st_int @ A @ A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_5802_atLeastAtMost__singleton,axiom,
    ! [A: nat] :
      ( ( set_or1269000886237332187st_nat @ A @ A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_5803_atLeastAtMost__singleton,axiom,
    ! [A: real] :
      ( ( set_or1222579329274155063t_real @ A @ A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% atLeastAtMost_singleton
thf(fact_5804_atLeastAtMost__singleton__iff,axiom,
    ! [A: $o,B: $o,C: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = ( insert_o @ C @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5805_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ C @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5806_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ C @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5807_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ C @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5808_mod__by__Suc__0,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_5809_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_5810_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W2: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
      = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_5811_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M2: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M2 ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_5812_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_5813_Gcd__fin_Oempty,axiom,
    ( ( semiri4258706085729940814in_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_fin.empty
thf(fact_5814_Gcd__fin_Oempty,axiom,
    ( ( semiri4256215615220890538in_int @ bot_bot_set_int )
    = zero_zero_int ) ).

% Gcd_fin.empty
thf(fact_5815_Gcd__fin_Oinfinite,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( semiri4258706085729940814in_nat @ A2 )
        = one_one_nat ) ) ).

% Gcd_fin.infinite
thf(fact_5816_Gcd__fin_Oinfinite,axiom,
    ! [A2: set_int] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( semiri4256215615220890538in_int @ A2 )
        = one_one_int ) ) ).

% Gcd_fin.infinite
thf(fact_5817_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_5818_Gcd__fin__eq__Gcd,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( semiri4258706085729940814in_nat @ A2 )
        = ( gcd_Gcd_nat @ A2 ) ) ) ).

% Gcd_fin_eq_Gcd
thf(fact_5819_Gcd__fin__eq__Gcd,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( semiri4256215615220890538in_int @ A2 )
        = ( gcd_Gcd_int @ A2 ) ) ) ).

% Gcd_fin_eq_Gcd
thf(fact_5820_count__notin,axiom,
    ! [X: $o,Xs: list_o] :
      ( ~ ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ( count_list_o @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_5821_count__notin,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
     => ( ( count_list_set_nat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_5822_count__notin,axiom,
    ! [X: set_nat_rat,Xs: list_set_nat_rat] :
      ( ~ ( member_set_nat_rat @ X @ ( set_set_nat_rat2 @ Xs ) )
     => ( ( count_6735058137522573441at_rat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_5823_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ( count_list_int @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_5824_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( count_list_VEBT_VEBT @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_5825_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( count_list_nat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_5826_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_5827_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_5828_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_5829_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_5830_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_5831_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
            = B ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_5832_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_5833_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_5834_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_5835_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_5836_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_5837_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_5838_inverse__eq__divide__numeral,axiom,
    ! [W2: num] :
      ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W2 ) )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W2 ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_5839_inverse__eq__divide__numeral,axiom,
    ! [W2: num] :
      ( ( inverse_inverse_real @ ( numeral_numeral_real @ W2 ) )
      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W2 ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_5840_inverse__eq__divide__numeral,axiom,
    ! [W2: num] :
      ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W2 ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W2 ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_5841_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
      = ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_5842_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
      = ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_5843_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_5844_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_5845_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5846_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5847_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5848_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5849_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5850_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5851_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5852_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5853_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5854_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5855_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5856_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5857_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5858_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5859_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5860_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5861_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_5862_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_5863_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).

% floor_less_numeral
thf(fact_5864_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_5865_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_5866_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_5867_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_5868_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_5869_nth__Cons__numeral,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,V: num] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
      = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_5870_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_5871_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_5872_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_5873_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
        = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).

% powr_numeral
thf(fact_5874_floor__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% floor_numeral_power
thf(fact_5875_floor__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% floor_numeral_power
thf(fact_5876_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% ceiling_numeral_power
thf(fact_5877_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_5878_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5879_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_5880_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5881_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_5882_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
            = B ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5883_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_5884_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
            = B ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5885_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5886_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_5887_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5888_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5889_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_5890_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5891_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_5892_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5893_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_5894_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5895_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_5896_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5897_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5898_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_5899_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5900_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_5901_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5902_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] :
      ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_5903_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] :
      ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_5904_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] :
      ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_5905_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_5906_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_5907_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_5908_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_5909_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_5910_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_5911_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_5912_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_5913_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_5914_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5915_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5916_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5917_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5918_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5919_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5920_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5921_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5922_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5923_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5924_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5925_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5926_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5927_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5928_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5929_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5930_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5931_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5932_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5933_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5934_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).

% numeral_less_floor
thf(fact_5935_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_5936_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% floor_le_numeral
thf(fact_5937_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_5938_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_5939_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_5940_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_5941_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_5942_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_5943_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_5944_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_5945_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_5946_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_5947_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_5948_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5949_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_5950_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_5951_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_5952_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5953_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_5954_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_5955_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_5956_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_5957_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_5958_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_5959_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5960_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_5961_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_5962_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_5963_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5964_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_5965_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_5966_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_5967_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_5968_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_5969_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% floor_le_neg_numeral
thf(fact_5970_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_5971_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_5972_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_5973_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_5974_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_5975_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_5976_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ 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_5977_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_5978_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_5979_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_5980_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ 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_5981_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_5982_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_5983_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_5984_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_5985_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ 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_5986_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_5987_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_5988_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_5989_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ 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_5990_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_5991_mod__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ M2 ) ).

% mod_less_eq_dividend
thf(fact_5992_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_5993_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_5994_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_5995_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_5996_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_z5237406670263579293d_enat
     != ( numera1916890842035813515d_enat @ N ) ) ).

% zero_neq_numeral
thf(fact_5997_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( numera6620942414471956472nteger @ N ) ) ).

% zero_neq_numeral
thf(fact_5998_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numera6620942414471956472nteger @ M2 )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5999_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_int @ M2 )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6000_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_real @ M2 )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6001_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_rat @ M2 )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6002_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numera6690914467698888265omplex @ M2 )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6003_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) )
     != ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6004_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
     != ( numeral_numeral_int @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6005_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) )
     != ( numeral_numeral_real @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6006_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) )
     != ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6007_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) )
     != ( numera6690914467698888265omplex @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6008_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_6009_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_6010_atLeastAtMost__singleton_H,axiom,
    ! [A: $o,B: $o] :
      ( ( A = B )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_6011_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_6012_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_6013_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_6014_mod__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_6015_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P6: nat,M2: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P6 )
       => ( ( ord_less_nat @ M2 @ P6 )
         => ( ! [N2: nat] :
                ( ( ord_less_nat @ N2 @ P6 )
               => ( ( P @ N2 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P6 ) ) ) )
           => ( P @ M2 ) ) ) ) ) ).

% mod_induct
thf(fact_6016_norm__ge__zero,axiom,
    ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).

% norm_ge_zero
thf(fact_6017_mod__less__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_6018_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N: nat] :
      ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
     => ( ! [M4: nat,N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ( P @ N2 @ ( modulo_modulo_nat @ M4 @ N2 ) )
             => ( P @ M4 @ N2 ) ) )
       => ( P @ M2 @ N ) ) ) ).

% gcd_nat_induct
thf(fact_6019_mod__Suc__le__divisor,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_6020_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_6021_zero__le__numeral,axiom,
    ! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% zero_le_numeral
thf(fact_6022_zero__le__numeral,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).

% zero_le_numeral
thf(fact_6023_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_6024_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_6025_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_6026_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_6027_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_le_zero
thf(fact_6028_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_le_zero
thf(fact_6029_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_6030_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_6031_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_6032_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_6033_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_6034_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_6035_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_6036_zero__less__numeral,axiom,
    ! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% zero_less_numeral
thf(fact_6037_zero__less__numeral,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).

% zero_less_numeral
thf(fact_6038_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_6039_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_6040_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_6041_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_6042_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_less_zero
thf(fact_6043_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_less_zero
thf(fact_6044_mod__eq__0D,axiom,
    ! [M2: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ D )
        = zero_zero_nat )
     => ? [Q5: nat] :
          ( M2
          = ( times_times_nat @ D @ Q5 ) ) ) ).

% mod_eq_0D
thf(fact_6045_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_6046_one__le__numeral,axiom,
    ! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% one_le_numeral
thf(fact_6047_one__le__numeral,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) ) ).

% one_le_numeral
thf(fact_6048_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_6049_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_6050_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_6051_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N4 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N4 ) @ N4 ) ) ) ) ).

% mod_if
thf(fact_6052_mod__geq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ M2 @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_6053_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_6054_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_6055_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_6056_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_6057_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).

% not_numeral_less_one
thf(fact_6058_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).

% not_numeral_less_one
thf(fact_6059_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6060_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6061_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6062_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6063_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6064_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6065_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6066_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6067_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6068_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6069_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6070_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6071_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6072_le__mod__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( modulo_modulo_nat @ M2 @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_6073_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6074_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6075_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6076_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6077_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6078_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6079_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6080_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6081_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_6082_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_6083_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_6084_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_6085_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_6086_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).

% one_plus_numeral_commute
thf(fact_6087_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_6088_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6089_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6090_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6091_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6092_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6093_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_6094_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_6095_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ N )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_6096_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_6097_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ N )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_6098_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_int @ A @ B )
          | ( ( ord_less_eq_set_int @ C @ A )
            & ( ord_less_eq_set_int @ B @ D )
            & ( ( ord_less_set_int @ C @ A )
              | ( ord_less_set_int @ B @ D ) ) ) )
        & ( ord_less_eq_set_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_6099_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_6100_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_6101_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_6102_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_6103_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_6104_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_6105_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_6106_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N: nat,Z: real] :
      ( ( ( power_power_real @ W2 @ N )
        = ( power_power_real @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V7735802525324610683m_real @ W2 )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_6107_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N: nat,Z: complex] :
      ( ( ( power_power_complex @ W2 @ N )
        = ( power_power_complex @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V1022390504157884413omplex @ W2 )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_6108_norm__mult__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_6109_norm__mult__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_6110_mod__le__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_6111_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
     => ( ( set_or1266510415728281911st_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M2 @ N ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_6112_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_6113_norm__power__ineq,axiom,
    ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).

% norm_power_ineq
thf(fact_6114_norm__power__ineq,axiom,
    ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).

% norm_power_ineq
thf(fact_6115_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_6116_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_6117_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_6118_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_6119_norm__triangle__le,axiom,
    ! [X: real,Y: real,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_le
thf(fact_6120_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_le
thf(fact_6121_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_6122_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_6123_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_le_diff
thf(fact_6124_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_le_diff
thf(fact_6125_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_6126_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_6127_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_6128_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_6129_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_6130_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_6131_div__less__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A2 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B2 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_6132_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_6133_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_6134_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_6135_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_6136_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6137_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6138_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_6139_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_6140_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_6141_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_6142_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_6143_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_6144_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6145_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_6146_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6147_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_6148_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_6149_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ( numeral_numeral_real @ W2 )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_6150_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_6151_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( numeral_numeral_real @ W2 ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_6152_norm__diff__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_6153_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_6154_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_6155_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_6156_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_6157_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).

% neg_numeral_le_one
thf(fact_6158_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_6159_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_6160_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_6161_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_6162_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_6163_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_6164_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_6165_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% neg_numeral_le_neg_one
thf(fact_6166_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_6167_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_6168_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_6169_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_le_neg_one
thf(fact_6170_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_6171_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_6172_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6173_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6174_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6175_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6176_mod__eq__nat1E,axiom,
    ! [M2: nat,Q4: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q4 )
        = ( modulo_modulo_nat @ N @ Q4 ) )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ~ ! [S3: nat] :
              ( M2
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q4 @ S3 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_6177_mod__eq__nat2E,axiom,
    ! [M2: nat,Q4: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q4 )
        = ( modulo_modulo_nat @ N @ Q4 ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ~ ! [S3: nat] :
              ( N
             != ( plus_plus_nat @ M2 @ ( times_times_nat @ Q4 @ S3 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_6178_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q5: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q5 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_6179_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_6180_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_6181_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).

% neg_numeral_less_one
thf(fact_6182_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_6183_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_6184_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_6185_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_6186_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_6187_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_6188_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_6189_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_less_neg_one
thf(fact_6190_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_6191_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6192_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6193_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6194_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6195_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6196_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6197_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6198_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6199_nonzero__norm__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
        = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).

% nonzero_norm_inverse
thf(fact_6200_nonzero__norm__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
        = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).

% nonzero_norm_inverse
thf(fact_6201_norm__exp,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).

% norm_exp
thf(fact_6202_norm__exp,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).

% norm_exp
thf(fact_6203_power__eq__1__iff,axiom,
    ! [W2: real,N: nat] :
      ( ( ( power_power_real @ W2 @ N )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W2 )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_6204_power__eq__1__iff,axiom,
    ! [W2: complex,N: nat] :
      ( ( ( power_power_complex @ W2 @ N )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W2 )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_6205_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_6206_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_6207_norm__sgn,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
          = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
          = one_one_real ) ) ) ).

% norm_sgn
thf(fact_6208_norm__sgn,axiom,
    ! [X: complex] :
      ( ( ( X = zero_zero_complex )
       => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
          = zero_zero_real ) )
      & ( ( X != zero_zero_complex )
       => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
          = one_one_real ) ) ) ).

% norm_sgn
thf(fact_6209_split__mod,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M2 ) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_6210_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_6211_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_6212_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_6213_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_6214_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6215_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_6216_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6217_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_6218_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_6219_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_6220_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% count_le_length
thf(fact_6221_count__le__length,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).

% count_le_length
thf(fact_6222_norm__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).

% norm_triangle_ineq3
thf(fact_6223_norm__triangle__ineq3,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% norm_triangle_ineq3
thf(fact_6224_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_6225_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_6226_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_6227_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_6228_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_6229_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_6230_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N ) ) @ M2 )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_6231_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_6232_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_6233_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_6234_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_6235_norm__inverse__le__norm,axiom,
    ! [R2: real,X: real] :
      ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ R2 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).

% norm_inverse_le_norm
thf(fact_6236_norm__inverse__le__norm,axiom,
    ! [R2: real,X: complex] :
      ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ R2 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).

% norm_inverse_le_norm
thf(fact_6237_CauchyD,axiom,
    ! [X5: nat > complex,E2: real] :
      ( ( topolo6517432010174082258omplex @ X5 )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ? [M9: nat] :
          ! [M: nat] :
            ( ( ord_less_eq_nat @ M9 @ M )
           => ! [N6: nat] :
                ( ( ord_less_eq_nat @ M9 @ N6 )
               => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X5 @ M ) @ ( X5 @ N6 ) ) ) @ E2 ) ) ) ) ) ).

% CauchyD
thf(fact_6238_CauchyD,axiom,
    ! [X5: nat > real,E2: real] :
      ( ( topolo4055970368930404560y_real @ X5 )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ? [M9: nat] :
          ! [M: nat] :
            ( ( ord_less_eq_nat @ M9 @ M )
           => ! [N6: nat] :
                ( ( ord_less_eq_nat @ M9 @ N6 )
               => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X5 @ M ) @ ( X5 @ N6 ) ) ) @ E2 ) ) ) ) ) ).

% CauchyD
thf(fact_6239_CauchyI,axiom,
    ! [X5: nat > complex] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ? [M10: nat] :
            ! [M4: nat] :
              ( ( ord_less_eq_nat @ M10 @ M4 )
             => ! [N2: nat] :
                  ( ( ord_less_eq_nat @ M10 @ N2 )
                 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) ) @ E ) ) ) )
     => ( topolo6517432010174082258omplex @ X5 ) ) ).

% CauchyI
thf(fact_6240_CauchyI,axiom,
    ! [X5: nat > real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ? [M10: nat] :
            ! [M4: nat] :
              ( ( ord_less_eq_nat @ M10 @ M4 )
             => ! [N2: nat] :
                  ( ( ord_less_eq_nat @ M10 @ N2 )
                 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) ) @ E ) ) ) )
     => ( topolo4055970368930404560y_real @ X5 ) ) ).

% CauchyI
thf(fact_6241_Cauchy__iff,axiom,
    ( topolo6517432010174082258omplex
    = ( ^ [X8: nat > complex] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [M8: nat] :
            ! [M3: nat] :
              ( ( ord_less_eq_nat @ M8 @ M3 )
             => ! [N4: nat] :
                  ( ( ord_less_eq_nat @ M8 @ N4 )
                 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) ) @ E3 ) ) ) ) ) ) ).

% Cauchy_iff
thf(fact_6242_Cauchy__iff,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [M8: nat] :
            ! [M3: nat] :
              ( ( ord_less_eq_nat @ M8 @ M3 )
             => ! [N4: nat] :
                  ( ( ord_less_eq_nat @ M8 @ N4 )
                 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) ) @ E3 ) ) ) ) ) ) ).

% Cauchy_iff
thf(fact_6243_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_6244_nth__rotate1,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) @ N )
        = ( nth_VEBT_VEBT @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_6245_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_6246_bset_I6_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B2 )
                 => ( X2
                   != ( plus_plus_int @ Xb @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X2 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_6247_bset_I8_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B2 )
                   => ( X2
                     != ( plus_plus_int @ Xb @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X2 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_6248_aset_I6_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A2 )
                   => ( X2
                     != ( minus_minus_int @ Xb @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X2 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_6249_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_6250_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
     => ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys2 ) @ N )
        = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_6251_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_6252_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys2 ) @ N )
        = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ 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_6253_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) @ N )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_6254_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys2 ) @ N )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ 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_6255_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_6256_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys2 ) @ N )
        = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_6257_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_6258_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_6259_enat__ord__number_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_6260_enat__ord__number_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_6261_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M4: nat] :
              ( ( ( some_nat @ M4 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_6262_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K4: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K4 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H ) ) @ K4 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_6263_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K4: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K4 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H ) ) @ K4 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_6264_bounded__linear__axioms__def,axiom,
    ( real_V7139242839884736329omplex
    = ( ^ [F5: complex > complex] :
        ? [K5: real] :
        ! [X3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F5 @ X3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 ) ) ) ) ).

% bounded_linear_axioms_def
thf(fact_6265_bounded__linear__axioms_Ointro,axiom,
    ! [F: complex > complex] :
      ( ? [K6: real] :
        ! [X4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X4 ) @ K6 ) )
     => ( real_V7139242839884736329omplex @ F ) ) ).

% bounded_linear_axioms.intro
thf(fact_6266_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% finite_atLeastAtMost
thf(fact_6267_pow__sum,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).

% pow_sum
thf(fact_6268_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% power_minus_is_div
thf(fact_6269_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% member_bound
thf(fact_6270_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M2: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X4 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_6271_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( some_nat @ M2 )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% misiz
thf(fact_6272_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_6273_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpyd
thf(fact_6274_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpypredd
thf(fact_6275_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_6276_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_6277_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_6278_semiring__norm_I71_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(71)
thf(fact_6279_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_6280_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
        | ( ord_less_nat @ Ma @ X ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_6281_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_6282_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_6283_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_6284_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_6285_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_6286_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H3: nat,L3: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L3 ) ) ) ).

% bit_concat_def
thf(fact_6287_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_6288_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_6289_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_6290_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_6291_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_6292_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_on7984719198319812577d_enat
        = ( numera1916890842035813515d_enat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_6293_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_integer
        = ( numera6620942414471956472nteger @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_6294_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6690914467698888265omplex @ N )
        = one_one_complex )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_6295_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_6296_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_6297_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_6298_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_6299_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera1916890842035813515d_enat @ N )
        = one_on7984719198319812577d_enat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_6300_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6620942414471956472nteger @ N )
        = one_one_Code_integer )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_6301_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_6302_semiring__norm_I69_J,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M2 ) @ one ) ).

% semiring_norm(69)
thf(fact_6303_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_6304_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_6305_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_6306_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6307_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_6308_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6309_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_6310_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_6311_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6312_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_6313_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6314_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_6315_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6316_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6317_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6318_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6319_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6320_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6321_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6322_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6323_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_6324_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_6325_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_6326_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_6327_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_6328_one__add__one,axiom,
    ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
    = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_6329_one__add__one,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_6330_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_6331_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_6332_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_6333_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% zero_eq_power2
thf(fact_6334_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% zero_eq_power2
thf(fact_6335_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_6336_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_6337_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_6338_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_6339_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_6340_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_6341_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_6342_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_6343_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_6344_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_6345_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_6346_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_6347_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_6348_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_6349_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_6350_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_6351_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_6352_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_6353_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_6354_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_6355_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_6356_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_6357_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_6358_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_6359_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_6360_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_6361_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_6362_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_6363_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_6364_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_6365_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_6366_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_6367_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_6368_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_6369_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_6370_one__div__two__eq__zero,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% one_div_two_eq_zero
thf(fact_6371_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_6372_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_6373_bits__1__div__2,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% bits_1_div_2
thf(fact_6374_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_6375_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_6376_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
thf(fact_6377_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_6378_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_6379_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_6380_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_6381_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_6382_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_6383_zero__less__power2,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_power2
thf(fact_6384_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_6385_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_6386_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_6387_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_6388_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_6389_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_6390_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_6391_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_6392_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_6393_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_6394_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_6395_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_6396_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_6397_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_6398_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_6399_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_6400_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_6401_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_6402_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_6403_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_6404_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_6405_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_6406_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_6407_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_6408_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_6409_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_6410_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_6411_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_6412_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_6413_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_6414_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_6415_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_6416_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_6417_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6418_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6419_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6420_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6421_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ one_one_complex )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6422_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_6423_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_6424_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_6425_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_6426_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_6427_add__self__mod__2,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_6428_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_6429_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_6430_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_6431_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_6432_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_6433_one__less__floor,axiom,
    ! [X: real] :
      ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_6434_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_6435_floor__le__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_6436_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_6437_mod2__gr__0,axiom,
    ! [M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_6438_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ N @ A2 )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).

% set_encode_insert
thf(fact_6439_card__2__iff_H,axiom,
    ! [S2: set_complex] :
      ( ( ( finite_card_complex @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: complex] :
            ( ( member_complex @ X3 @ S2 )
            & ? [Y2: complex] :
                ( ( member_complex @ Y2 @ S2 )
                & ( X3 != Y2 )
                & ! [Z2: complex] :
                    ( ( member_complex @ Z2 @ S2 )
                   => ( ( Z2 = X3 )
                      | ( Z2 = Y2 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_6440_card__2__iff_H,axiom,
    ! [S2: set_list_nat] :
      ( ( ( finite_card_list_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: list_nat] :
            ( ( member_list_nat @ X3 @ S2 )
            & ? [Y2: list_nat] :
                ( ( member_list_nat @ Y2 @ S2 )
                & ( X3 != Y2 )
                & ! [Z2: list_nat] :
                    ( ( member_list_nat @ Z2 @ S2 )
                   => ( ( Z2 = X3 )
                      | ( Z2 = Y2 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_6441_card__2__iff_H,axiom,
    ! [S2: set_set_nat] :
      ( ( ( finite_card_set_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ S2 )
            & ? [Y2: set_nat] :
                ( ( member_set_nat @ Y2 @ S2 )
                & ( X3 != Y2 )
                & ! [Z2: set_nat] :
                    ( ( member_set_nat @ Z2 @ S2 )
                   => ( ( Z2 = X3 )
                      | ( Z2 = Y2 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_6442_card__2__iff_H,axiom,
    ! [S2: set_nat] :
      ( ( ( finite_card_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ S2 )
            & ? [Y2: nat] :
                ( ( member_nat @ Y2 @ S2 )
                & ( X3 != Y2 )
                & ! [Z2: nat] :
                    ( ( member_nat @ Z2 @ S2 )
                   => ( ( Z2 = X3 )
                      | ( Z2 = Y2 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_6443_card__2__iff_H,axiom,
    ! [S2: set_int] :
      ( ( ( finite_card_int @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: int] :
            ( ( member_int @ X3 @ S2 )
            & ? [Y2: int] :
                ( ( member_int @ Y2 @ S2 )
                & ( X3 != Y2 )
                & ! [Z2: int] :
                    ( ( member_int @ Z2 @ S2 )
                   => ( ( Z2 = X3 )
                      | ( Z2 = Y2 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_6444_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_6445_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_6446_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_6447_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_6448_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_6449_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_6450_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_6451_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_6452_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_6453_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_6454_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_6455_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_6456_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_6457_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_6458_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_6459_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_6460_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_6461_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_6462_power2__nat__le__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_6463_power2__nat__le__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_6464_self__le__ge2__pow,axiom,
    ! [K: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ M2 @ ( power_power_nat @ K @ M2 ) ) ) ).

% self_le_ge2_pow
thf(fact_6465_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_6466_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_6467_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_6468_zero__le__power2,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_6469_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_6470_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_6471_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_6472_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_6473_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_6474_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_6475_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_6476_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_6477_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_6478_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_6479_power2__less__0,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).

% power2_less_0
thf(fact_6480_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_6481_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_6482_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_6483_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_6484_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_6485_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_6486_mult__2,axiom,
    ! [Z: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z )
      = ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).

% mult_2
thf(fact_6487_mult__2,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Z )
      = ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).

% mult_2
thf(fact_6488_mult__2__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6489_mult__2__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6490_mult__2__right,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6491_mult__2__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6492_mult__2__right,axiom,
    ! [Z: extended_enat] :
      ( ( times_7803423173614009249d_enat @ Z @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6493_mult__2__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6494_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_6495_left__add__twice,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_6496_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_6497_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_6498_left__add__twice,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ A @ B ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_6499_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_6500_field__sum__of__halves,axiom,
    ! [X: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_6501_field__sum__of__halves,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_6502_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_6503_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( A = one_one_real )
        | ( A
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% power2_eq_1_iff
thf(fact_6504_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_6505_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
      = ( ( A = one_one_complex )
        | ( A
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% power2_eq_1_iff
thf(fact_6506_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
      = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6507_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6508_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
      = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6509_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_6510_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_6511_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_6512_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_6513_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_6514_card__2__iff,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] :
            ( ( S2
              = ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ Y2 @ bot_bo2099793752762293965at_nat ) ) )
            & ( X3 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_6515_card__2__iff,axiom,
    ! [S2: set_complex] :
      ( ( ( finite_card_complex @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: complex,Y2: complex] :
            ( ( S2
              = ( insert_complex @ X3 @ ( insert_complex @ Y2 @ bot_bot_set_complex ) ) )
            & ( X3 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_6516_card__2__iff,axiom,
    ! [S2: set_list_nat] :
      ( ( ( finite_card_list_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: list_nat,Y2: list_nat] :
            ( ( S2
              = ( insert_list_nat @ X3 @ ( insert_list_nat @ Y2 @ bot_bot_set_list_nat ) ) )
            & ( X3 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_6517_card__2__iff,axiom,
    ! [S2: set_set_nat] :
      ( ( ( finite_card_set_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: set_nat,Y2: set_nat] :
            ( ( S2
              = ( insert_set_nat @ X3 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) )
            & ( X3 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_6518_card__2__iff,axiom,
    ! [S2: set_real] :
      ( ( ( finite_card_real @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: real,Y2: real] :
            ( ( S2
              = ( insert_real @ X3 @ ( insert_real @ Y2 @ bot_bot_set_real ) ) )
            & ( X3 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_6519_card__2__iff,axiom,
    ! [S2: set_o] :
      ( ( ( finite_card_o @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: $o,Y2: $o] :
            ( ( S2
              = ( insert_o @ X3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
            & ( X3 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_6520_card__2__iff,axiom,
    ! [S2: set_nat] :
      ( ( ( finite_card_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: nat,Y2: nat] :
            ( ( S2
              = ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
            & ( X3 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_6521_card__2__iff,axiom,
    ! [S2: set_int] :
      ( ( ( finite_card_int @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: int,Y2: int] :
            ( ( S2
              = ( insert_int @ X3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
            & ( X3 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_6522_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_6523_nat__induct2,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct2
thf(fact_6524_square__fact__le__2__fact,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% square_fact_le_2_fact
thf(fact_6525_two__realpow__ge__one,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% two_realpow_ge_one
thf(fact_6526_diff__le__diff__pow,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ ( power_power_nat @ K @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_6527_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_6528_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_6529_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_real @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_6530_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_6531_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_nat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_6532_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_int @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_6533_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_6534_half__gt__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_6535_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_6536_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% half_gt_zero_iff
thf(fact_6537_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_6538_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_6539_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_6540_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_6541_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_6542_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_6543_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_6544_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_6545_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_6546_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_6547_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_6548_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_6549_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_6550_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_6551_square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_6552_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_6553_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_6554_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_6555_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_6556_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_6557_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_6558_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6559_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6560_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6561_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_6562_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_6563_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
thf(fact_6564_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_6565_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_6566_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
thf(fact_6567_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_6568_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_6569_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_6570_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: 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 @ M2 ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_6571_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: 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 @ M2 ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_6572_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) )
       != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_6573_abs__square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_6574_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_6575_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_6576_abs__square__less__1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_6577_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_6578_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_6579_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less
thf(fact_6580_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less
thf(fact_6581_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_6582_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_6583_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_6584_square__norm__one,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_6585_square__norm__one,axiom,
    ! [X: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_6586_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_6587_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_6588_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_Bit0
thf(fact_6589_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_6590_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_6591_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).

% numeral_Bit0
thf(fact_6592_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_Bit0
thf(fact_6593_exp__half__le2,axiom,
    ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% exp_half_le2
thf(fact_6594_exp__plus__inverse__exp,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_6595_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6596_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6597_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6598_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6599_mult__numeral__1__right,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6600_mult__numeral__1__right,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6601_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6602_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6603_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6604_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6605_mult__numeral__1,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6606_mult__numeral__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6607_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_6608_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_6609_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_6610_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_6611_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_6612_numeral__One,axiom,
    ( ( numera1916890842035813515d_enat @ one )
    = one_on7984719198319812577d_enat ) ).

% numeral_One
thf(fact_6613_numeral__One,axiom,
    ( ( numera6620942414471956472nteger @ one )
    = one_one_Code_integer ) ).

% numeral_One
thf(fact_6614_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_6615_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_6616_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_6617_inverse__numeral__1,axiom,
    ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
    = ( numeral_numeral_real @ one ) ) ).

% inverse_numeral_1
thf(fact_6618_inverse__numeral__1,axiom,
    ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
    = ( numeral_numeral_rat @ one ) ) ).

% inverse_numeral_1
thf(fact_6619_sum__squares__bound,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_6620_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_6621_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_6622_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_6623_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_6624_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_6625_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_6626_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_6627_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_6628_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_6629_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_6630_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_6631_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_6632_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_6633_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_6634_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_6635_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_6636_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_6637_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 )
       => ? [N2: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_6638_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 )
       => ? [N2: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_6639_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_6640_exp__bound__half,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_6641_exp__bound__half,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_6642_less__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% less_log2_of_power
thf(fact_6643_le__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% le_log2_of_power
thf(fact_6644_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_6645_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_6646_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_6647_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_6648_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( 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 @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_6649_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( 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 @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_6650_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_6651_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_6652_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_6653_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_6654_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( set_or1269000886237332187st_nat @ M2 @ N )
        = ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_6655_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) )
        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_6656_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_6657_num_Osize_I5_J,axiom,
    ! [X23: num] :
      ( ( size_size_num @ ( bit0 @ X23 ) )
      = ( plus_plus_nat @ ( size_size_num @ X23 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_6658_log2__of__power__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_6659_exp__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_6660_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_6661_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_6662_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_6663_real__le__x__sinh,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_6664_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_6665_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_6666_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_6667_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_6668_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_6669_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_6670_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_6671_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_6672_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_6673_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_6674_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_6675_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_6676_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_6677_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_6678_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_6679_real__le__abs__sinh,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_6680_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_6681_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_6682_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_6683_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X @ Y ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_6684_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X @ Y ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_6685_mod__double__modulus,axiom,
    ! [M2: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo364778990260209775nteger @ X @ M2 ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_6686_mod__double__modulus,axiom,
    ! [M2: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_nat @ X @ M2 ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_6687_mod__double__modulus,axiom,
    ! [M2: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_int @ X @ M2 ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_6688_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_6689_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_6690_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_6691_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_6692_log2__of__power__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_6693_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_6694_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_6695_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_6696_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_6697_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_6698_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_6699_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_6700_floor__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_6701_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_6702_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X4 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2 = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_6703_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% fact_double
thf(fact_6704_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_6705_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_double
thf(fact_6706_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_6707_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X4 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_6708_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: 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 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_6709_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_6710_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_6711_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_6712_pochhammer__double,axiom,
    ! [Z: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_6713_pochhammer__double,axiom,
    ! [Z: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_6714_pochhammer__double,axiom,
    ! [Z: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_6715_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_6716_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_6717_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_6718_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_6719_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_6720_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_6721_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_6722_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_6723_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q4: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_6724_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q4: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_6725_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q4: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_6726_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
         => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6727_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
         => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6728_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X: int] :
      ( ! [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
         => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6729_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_6730_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_6731_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_6732_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_6733_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_6734_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_6735_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_6736_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_6737_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X3: nat,N4: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% high_def
thf(fact_6738_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% high_bound_aux
thf(fact_6739_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = Y ) ) ).

% high_inv
thf(fact_6740_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_6741_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_6742_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_6743_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_6744_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_6745_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_6746_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_6747_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).

% complex_mod_triangle_ineq2
thf(fact_6748_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( N
          = ( suc @ ( suc @ Va2 ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_6749_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_6750_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X = Mi )
          | ( X = Ma )
          | ( ( ord_less_nat @ X @ Ma )
            & ( ord_less_nat @ Mi @ X )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_6751_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_6752_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_6753_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_6754_bit__split__inv,axiom,
    ! [X: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
      = X ) ).

% bit_split_inv
thf(fact_6755_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X3: nat,N4: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% low_def
thf(fact_6756_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = X ) ) ).

% low_inv
thf(fact_6757_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_6758_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X4 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2 = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X4: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_6759_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X4 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X4: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_6760_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A5: $o,B5: $o] :
              ( A12
              = ( vEBT_Leaf @ A5 @ B5 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_invar_vebt @ X2 @ N2 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                     => ( ( M4 = N2 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N2 @ M4 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_invar_vebt @ X2 @ N2 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                       => ( ( M4
                            = ( suc @ N2 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N2 @ M4 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_invar_vebt @ X2 @ N2 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                         => ( ( M4 = N2 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N2 @ M4 ) )
                             => ( ! [I3: nat] :
                                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I3: nat] :
                                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                    = I3 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                & ! [X2: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
                                                        = I3 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X2 )
                                                      & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ( vEBT_invar_vebt @ X2 @ N2 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                           => ( ( M4
                                = ( suc @ N2 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N2 @ M4 ) )
                               => ( ! [I3: nat] :
                                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I3: nat] :
                                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                      = I3 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                  & ! [X2: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
                                                          = I3 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X2 )
                                                        & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_6761_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A13: vEBT_VEBT,A24: nat] :
          ( ( ? [A4: $o,B4: $o] :
                ( A13
                = ( vEBT_Leaf @ A4 @ B4 ) )
            & ( A24
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
              ( ( A13
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A24 @ TreeList3 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X3 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N4 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
              & ( A24
                = ( plus_plus_nat @ N4 @ N4 ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
              ( ( A13
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A24 @ TreeList3 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X3 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
              & ( A24
                = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A13
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList3 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X3 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N4 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
              & ( A24
                = ( plus_plus_nat @ N4 @ N4 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A24 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
                          = I4 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
                      & ! [X3: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X3 @ N4 )
                              = I4 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A13
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList3 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X3 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
              & ( A24
                = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A24 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
                          = I4 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
                      & ! [X3: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X3 @ N4 )
                              = I4 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_6762_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N4: nat,TreeList3: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X3 @ N4 ) ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) ) ) ).

% in_children_def
thf(fact_6763_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_6764_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_6765_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_6766_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_6767_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_6768_finite__Collect__conjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X3: set_nat] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6769_finite__Collect__conjI,axiom,
    ! [P: set_nat_rat > $o,Q: set_nat_rat > $o] :
      ( ( ( finite6430367030675640852at_rat @ ( collect_set_nat_rat @ P ) )
        | ( finite6430367030675640852at_rat @ ( collect_set_nat_rat @ Q ) ) )
     => ( finite6430367030675640852at_rat
        @ ( collect_set_nat_rat
          @ ^ [X3: set_nat_rat] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6770_finite__Collect__conjI,axiom,
    ! [P: ( nat > rat ) > $o,Q: ( nat > rat ) > $o] :
      ( ( ( finite7830837933032798814at_rat @ ( collect_nat_rat @ P ) )
        | ( finite7830837933032798814at_rat @ ( collect_nat_rat @ Q ) ) )
     => ( finite7830837933032798814at_rat
        @ ( collect_nat_rat
          @ ^ [X3: nat > rat] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6771_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6772_finite__Collect__conjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ( finite_finite_int @ ( collect_int @ P ) )
        | ( finite_finite_int @ ( collect_int @ Q ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6773_finite__Collect__conjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X3: complex] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6774_finite__Collect__conjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X3: product_prod_nat_nat] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6775_finite__Collect__conjI,axiom,
    ! [P: extended_enat > $o,Q: extended_enat > $o] :
      ( ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
        | ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) )
     => ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [X3: extended_enat] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6776_finite__Collect__disjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X3: set_nat] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6777_finite__Collect__disjI,axiom,
    ! [P: set_nat_rat > $o,Q: set_nat_rat > $o] :
      ( ( finite6430367030675640852at_rat
        @ ( collect_set_nat_rat
          @ ^ [X3: set_nat_rat] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite6430367030675640852at_rat @ ( collect_set_nat_rat @ P ) )
        & ( finite6430367030675640852at_rat @ ( collect_set_nat_rat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6778_finite__Collect__disjI,axiom,
    ! [P: ( nat > rat ) > $o,Q: ( nat > rat ) > $o] :
      ( ( finite7830837933032798814at_rat
        @ ( collect_nat_rat
          @ ^ [X3: nat > rat] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite7830837933032798814at_rat @ ( collect_nat_rat @ P ) )
        & ( finite7830837933032798814at_rat @ ( collect_nat_rat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6779_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6780_finite__Collect__disjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P ) )
        & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6781_finite__Collect__disjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X3: complex] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6782_finite__Collect__disjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X3: product_prod_nat_nat] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6783_finite__Collect__disjI,axiom,
    ! [P: extended_enat > $o,Q: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [X3: extended_enat] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
        & ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6784_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y2: nat] :
                ( ( vEBT_vebt_member @ T @ Y2 )
                & ( ord_less_nat @ Y2 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_6785_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y2: nat] :
                ( ( vEBT_vebt_member @ T @ Y2 )
                & ( ord_less_nat @ X @ Y2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_6786_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_6787_finite__Collect__subsets,axiom,
    ! [A2: set_nat_rat] :
      ( ( finite7830837933032798814at_rat @ A2 )
     => ( finite6430367030675640852at_rat
        @ ( collect_set_nat_rat
          @ ^ [B6: set_nat_rat] : ( ord_le2679597024174929757at_rat @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6788_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6789_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6790_finite__Collect__subsets,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [B6: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6791_finite__Collect__subsets,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite5468666774076196335d_enat
        @ ( collec2260605976452661553d_enat
          @ ^ [B6: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6792_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6793_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X3: product_prod_nat_nat] : ( X3 = A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_6794_singleton__conv,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ^ [X3: set_nat] : ( X3 = A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_6795_singleton__conv,axiom,
    ! [A: set_nat_rat] :
      ( ( collect_set_nat_rat
        @ ^ [X3: set_nat_rat] : ( X3 = A ) )
      = ( insert_set_nat_rat @ A @ bot_bo6797373522285170759at_rat ) ) ).

% singleton_conv
thf(fact_6796_singleton__conv,axiom,
    ! [A: nat > rat] :
      ( ( collect_nat_rat
        @ ^ [X3: nat > rat] : ( X3 = A ) )
      = ( insert_nat_rat @ A @ bot_bot_set_nat_rat ) ) ).

% singleton_conv
thf(fact_6797_singleton__conv,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ^ [X3: real] : ( X3 = A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_6798_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X3: $o] : ( X3 = A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_6799_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( X3 = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_6800_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X3: int] : ( X3 = A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_6801_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y5: product_prod_nat_nat,Z4: product_prod_nat_nat] : ( Y5 = Z4 )
          @ A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_6802_singleton__conv2,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ( ^ [Y5: set_nat,Z4: set_nat] : ( Y5 = Z4 )
          @ A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv2
thf(fact_6803_singleton__conv2,axiom,
    ! [A: set_nat_rat] :
      ( ( collect_set_nat_rat
        @ ( ^ [Y5: set_nat_rat,Z4: set_nat_rat] : ( Y5 = Z4 )
          @ A ) )
      = ( insert_set_nat_rat @ A @ bot_bo6797373522285170759at_rat ) ) ).

% singleton_conv2
thf(fact_6804_singleton__conv2,axiom,
    ! [A: nat > rat] :
      ( ( collect_nat_rat
        @ ( ^ [Y5: nat > rat,Z4: nat > rat] : ( Y5 = Z4 )
          @ A ) )
      = ( insert_nat_rat @ A @ bot_bot_set_nat_rat ) ) ).

% singleton_conv2
thf(fact_6805_singleton__conv2,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 )
          @ A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_6806_singleton__conv2,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ( ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
          @ A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_6807_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_6808_singleton__conv2,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
          @ A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_6809_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_6810_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_6811_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_6812_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_6813_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_6814_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( list_update_nat @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_6815_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_6816_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_6817_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_6818_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_6819_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_6820_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_6821_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
          = ( set_int2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_6822_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_6823_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_6824_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                      @ ( vEBT_vebt_delete @ Summary @ H ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_6825_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_6826_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary @ H ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_6827_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_x_mia
thf(fact_6828_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                     => ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( Sn
                            = ( vEBT_vebt_delete @ Summary @ H ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_6829_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary @ H ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_6830_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                      @ ( vEBT_Node
                        @ ( some_P7363390416028606310at_nat
                          @ ( product_Pair_nat_nat @ Xn
                            @ ( if_nat @ ( Xn = Ma )
                              @ ( if_nat
                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
                                  = none_nat )
                                @ Xn
                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                        @ ( vEBT_vebt_delete @ Summary @ H ) )
                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_6831_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_6832_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_6833_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_6834_bot__empty__eq2,axiom,
    ( bot_bo4898103413517107610_nat_o
    = ( ^ [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y2 ) @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6835_bot__empty__eq2,axiom,
    ( bot_bo3364206721330744218_nat_o
    = ( ^ [X3: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y2 ) @ bot_bo4948859079157340979at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6836_bot__empty__eq2,axiom,
    ( bot_bo394778441745866138_nat_o
    = ( ^ [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y2 ) @ bot_bo228742789529271731at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6837_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6838_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_6839_Collect__conv__if,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_6840_Collect__conv__if,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X3: set_nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X3: set_nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_6841_Collect__conv__if,axiom,
    ! [P: set_nat_rat > $o,A: set_nat_rat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat_rat
            @ ^ [X3: set_nat_rat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert_set_nat_rat @ A @ bot_bo6797373522285170759at_rat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat_rat
            @ ^ [X3: set_nat_rat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bo6797373522285170759at_rat ) ) ) ).

% Collect_conv_if
thf(fact_6842_Collect__conv__if,axiom,
    ! [P: ( nat > rat ) > $o,A: nat > rat] :
      ( ( ( P @ A )
       => ( ( collect_nat_rat
            @ ^ [X3: nat > rat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert_nat_rat @ A @ bot_bot_set_nat_rat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat_rat
            @ ^ [X3: nat > rat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bot_set_nat_rat ) ) ) ).

% Collect_conv_if
thf(fact_6843_Collect__conv__if,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X3: real] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X3: real] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_6844_Collect__conv__if,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_6845_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_6846_Collect__conv__if,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X3: int] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X3: int] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_6847_Collect__conv__if2,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6848_Collect__conv__if2,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X3: set_nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X3: set_nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6849_Collect__conv__if2,axiom,
    ! [P: set_nat_rat > $o,A: set_nat_rat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat_rat
            @ ^ [X3: set_nat_rat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert_set_nat_rat @ A @ bot_bo6797373522285170759at_rat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat_rat
            @ ^ [X3: set_nat_rat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bo6797373522285170759at_rat ) ) ) ).

% Collect_conv_if2
thf(fact_6850_Collect__conv__if2,axiom,
    ! [P: ( nat > rat ) > $o,A: nat > rat] :
      ( ( ( P @ A )
       => ( ( collect_nat_rat
            @ ^ [X3: nat > rat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert_nat_rat @ A @ bot_bot_set_nat_rat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat_rat
            @ ^ [X3: nat > rat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_nat_rat ) ) ) ).

% Collect_conv_if2
thf(fact_6851_Collect__conv__if2,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X3: real] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X3: real] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_6852_Collect__conv__if2,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_6853_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6854_Collect__conv__if2,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X3: int] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X3: int] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_6855_empty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X3: set_nat] : $false ) ) ).

% empty_def
thf(fact_6856_empty__def,axiom,
    ( bot_bo6797373522285170759at_rat
    = ( collect_set_nat_rat
      @ ^ [X3: set_nat_rat] : $false ) ) ).

% empty_def
thf(fact_6857_empty__def,axiom,
    ( bot_bot_set_nat_rat
    = ( collect_nat_rat
      @ ^ [X3: nat > rat] : $false ) ) ).

% empty_def
thf(fact_6858_empty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X3: real] : $false ) ) ).

% empty_def
thf(fact_6859_empty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X3: $o] : $false ) ) ).

% empty_def
thf(fact_6860_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X3: nat] : $false ) ) ).

% empty_def
thf(fact_6861_empty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X3: int] : $false ) ) ).

% empty_def
thf(fact_6862_lambda__one,axiom,
    ( ( ^ [X3: complex] : X3 )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_6863_lambda__one,axiom,
    ( ( ^ [X3: real] : X3 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_6864_lambda__one,axiom,
    ( ( ^ [X3: rat] : X3 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_6865_lambda__one,axiom,
    ( ( ^ [X3: nat] : X3 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_6866_lambda__one,axiom,
    ( ( ^ [X3: int] : X3 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_6867_pred__subset__eq,axiom,
    ! [R: set_o,S2: set_o] :
      ( ( ord_less_eq_o_o
        @ ^ [X3: $o] : ( member_o @ X3 @ R )
        @ ^ [X3: $o] : ( member_o @ X3 @ S2 ) )
      = ( ord_less_eq_set_o @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_6868_pred__subset__eq,axiom,
    ! [R: set_set_nat,S2: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R )
        @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S2 ) )
      = ( ord_le6893508408891458716et_nat @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_6869_pred__subset__eq,axiom,
    ! [R: set_set_nat_rat,S2: set_set_nat_rat] :
      ( ( ord_le4100815579384348210_rat_o
        @ ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ R )
        @ ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ S2 ) )
      = ( ord_le4375437777232675859at_rat @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_6870_pred__subset__eq,axiom,
    ! [R: set_nat,S2: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X3: nat] : ( member_nat @ X3 @ R )
        @ ^ [X3: nat] : ( member_nat @ X3 @ S2 ) )
      = ( ord_less_eq_set_nat @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_6871_pred__subset__eq,axiom,
    ! [R: set_int,S2: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X3: int] : ( member_int @ X3 @ R )
        @ ^ [X3: int] : ( member_int @ X3 @ S2 ) )
      = ( ord_less_eq_set_int @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_6872_Collect__subset,axiom,
    ! [A2: set_o,P: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6873_Collect__subset,axiom,
    ! [A2: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6874_Collect__subset,axiom,
    ! [A2: set_set_nat_rat,P: set_nat_rat > $o] :
      ( ord_le4375437777232675859at_rat
      @ ( collect_set_nat_rat
        @ ^ [X3: set_nat_rat] :
            ( ( member_set_nat_rat @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6875_Collect__subset,axiom,
    ! [A2: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6876_Collect__subset,axiom,
    ! [A2: set_nat_rat,P: ( nat > rat ) > $o] :
      ( ord_le2679597024174929757at_rat
      @ ( collect_nat_rat
        @ ^ [X3: nat > rat] :
            ( ( member_nat_rat @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6877_Collect__subset,axiom,
    ! [A2: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6878_less__eq__set__def,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( ord_less_eq_o_o
          @ ^ [X3: $o] : ( member_o @ X3 @ A6 )
          @ ^ [X3: $o] : ( member_o @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_6879_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_6880_less__eq__set__def,axiom,
    ( ord_le4375437777232675859at_rat
    = ( ^ [A6: set_set_nat_rat,B6: set_set_nat_rat] :
          ( ord_le4100815579384348210_rat_o
          @ ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ A6 )
          @ ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_6881_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
          @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_6882_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X3: int] : ( member_int @ X3 @ A6 )
          @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_6883_Collect__restrict,axiom,
    ! [X5: set_o,P: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o @ X3 @ X5 )
            & ( P @ X3 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_6884_Collect__restrict,axiom,
    ! [X5: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X3: set_nat] :
            ( ( member_set_nat @ X3 @ X5 )
            & ( P @ X3 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_6885_Collect__restrict,axiom,
    ! [X5: set_set_nat_rat,P: set_nat_rat > $o] :
      ( ord_le4375437777232675859at_rat
      @ ( collect_set_nat_rat
        @ ^ [X3: set_nat_rat] :
            ( ( member_set_nat_rat @ X3 @ X5 )
            & ( P @ X3 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_6886_Collect__restrict,axiom,
    ! [X5: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X3: nat] :
            ( ( member_nat @ X3 @ X5 )
            & ( P @ X3 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_6887_Collect__restrict,axiom,
    ! [X5: set_nat_rat,P: ( nat > rat ) > $o] :
      ( ord_le2679597024174929757at_rat
      @ ( collect_nat_rat
        @ ^ [X3: nat > rat] :
            ( ( member_nat_rat @ X3 @ X5 )
            & ( P @ X3 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_6888_Collect__restrict,axiom,
    ! [X5: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X3: int] :
            ( ( member_int @ X3 @ X5 )
            & ( P @ X3 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_6889_prop__restrict,axiom,
    ! [X: $o,Z7: set_o,X5: set_o,P: $o > $o] :
      ( ( member_o @ X @ Z7 )
     => ( ( ord_less_eq_set_o @ Z7
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ X5 )
                & ( P @ X3 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_6890_prop__restrict,axiom,
    ! [X: set_nat,Z7: set_set_nat,X5: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X @ Z7 )
     => ( ( ord_le6893508408891458716et_nat @ Z7
          @ ( collect_set_nat
            @ ^ [X3: set_nat] :
                ( ( member_set_nat @ X3 @ X5 )
                & ( P @ X3 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_6891_prop__restrict,axiom,
    ! [X: set_nat_rat,Z7: set_set_nat_rat,X5: set_set_nat_rat,P: set_nat_rat > $o] :
      ( ( member_set_nat_rat @ X @ Z7 )
     => ( ( ord_le4375437777232675859at_rat @ Z7
          @ ( collect_set_nat_rat
            @ ^ [X3: set_nat_rat] :
                ( ( member_set_nat_rat @ X3 @ X5 )
                & ( P @ X3 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_6892_prop__restrict,axiom,
    ! [X: nat,Z7: set_nat,X5: set_nat,P: nat > $o] :
      ( ( member_nat @ X @ Z7 )
     => ( ( ord_less_eq_set_nat @ Z7
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ X5 )
                & ( P @ X3 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_6893_prop__restrict,axiom,
    ! [X: nat > rat,Z7: set_nat_rat,X5: set_nat_rat,P: ( nat > rat ) > $o] :
      ( ( member_nat_rat @ X @ Z7 )
     => ( ( ord_le2679597024174929757at_rat @ Z7
          @ ( collect_nat_rat
            @ ^ [X3: nat > rat] :
                ( ( member_nat_rat @ X3 @ X5 )
                & ( P @ X3 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_6894_prop__restrict,axiom,
    ! [X: int,Z7: set_int,X5: set_int,P: int > $o] :
      ( ( member_int @ X @ Z7 )
     => ( ( ord_less_eq_set_int @ Z7
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ X5 )
                & ( P @ X3 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_6895_less__set__def,axiom,
    ( ord_less_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( ord_less_o_o
          @ ^ [X3: $o] : ( member_o @ X3 @ A6 )
          @ ^ [X3: $o] : ( member_o @ X3 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_6896_less__set__def,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( ord_less_set_nat_o
          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_6897_less__set__def,axiom,
    ( ord_le1311537459589289991at_rat
    = ( ^ [A6: set_set_nat_rat,B6: set_set_nat_rat] :
          ( ord_le6823063569548456766_rat_o
          @ ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ A6 )
          @ ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_6898_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_nat_o
          @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
          @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_6899_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_int_o
          @ ^ [X3: int] : ( member_int @ X3 @ A6 )
          @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_6900_insert__compr,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X3: product_prod_nat_nat] :
              ( ( X3 = A4 )
              | ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6901_insert__compr,axiom,
    ( insert_real
    = ( ^ [A4: real,B6: set_real] :
          ( collect_real
          @ ^ [X3: real] :
              ( ( X3 = A4 )
              | ( member_real @ X3 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6902_insert__compr,axiom,
    ( insert_o
    = ( ^ [A4: $o,B6: set_o] :
          ( collect_o
          @ ^ [X3: $o] :
              ( ( X3 = A4 )
              | ( member_o @ X3 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6903_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A4: set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X3: set_nat] :
              ( ( X3 = A4 )
              | ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6904_insert__compr,axiom,
    ( insert_set_nat_rat
    = ( ^ [A4: set_nat_rat,B6: set_set_nat_rat] :
          ( collect_set_nat_rat
          @ ^ [X3: set_nat_rat] :
              ( ( X3 = A4 )
              | ( member_set_nat_rat @ X3 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6905_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A4: nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ( ( X3 = A4 )
              | ( member_nat @ X3 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6906_insert__compr,axiom,
    ( insert_int
    = ( ^ [A4: int,B6: set_int] :
          ( collect_int
          @ ^ [X3: int] :
              ( ( X3 = A4 )
              | ( member_int @ X3 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6907_insert__compr,axiom,
    ( insert_nat_rat
    = ( ^ [A4: nat > rat,B6: set_nat_rat] :
          ( collect_nat_rat
          @ ^ [X3: nat > rat] :
              ( ( X3 = A4 )
              | ( member_nat_rat @ X3 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_6908_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( collec3392354462482085612at_nat
        @ ^ [U2: product_prod_nat_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6909_insert__Collect,axiom,
    ! [A: real,P: real > $o] :
      ( ( insert_real @ A @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U2: real] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6910_insert__Collect,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( insert_o @ A @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U2: $o] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6911_insert__Collect,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( collect_set_nat
        @ ^ [U2: set_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6912_insert__Collect,axiom,
    ! [A: set_nat_rat,P: set_nat_rat > $o] :
      ( ( insert_set_nat_rat @ A @ ( collect_set_nat_rat @ P ) )
      = ( collect_set_nat_rat
        @ ^ [U2: set_nat_rat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6913_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6914_insert__Collect,axiom,
    ! [A: int,P: int > $o] :
      ( ( insert_int @ A @ ( collect_int @ P ) )
      = ( collect_int
        @ ^ [U2: int] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6915_insert__Collect,axiom,
    ! [A: nat > rat,P: ( nat > rat ) > $o] :
      ( ( insert_nat_rat @ A @ ( collect_nat_rat @ P ) )
      = ( collect_nat_rat
        @ ^ [U2: nat > rat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6916_pred__subset__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y2 ) @ R )
        @ ^ [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y2 ) @ S2 ) )
      = ( ord_le3000389064537975527at_nat @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_6917_pred__subset__eq2,axiom,
    ! [R: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ( ord_le3072208448688395470_nat_o
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y2 ) @ R )
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y2 ) @ S2 ) )
      = ( ord_le5997549366648089703at_nat @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_6918_pred__subset__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y2 ) @ R )
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y2 ) @ S2 ) )
      = ( ord_le1268244103169919719at_nat @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_6919_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R )
        @ ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ S2 ) )
      = ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_6920_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R )
        @ ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ S2 ) )
      = ( ord_le2843351958646193337nt_int @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_6921_uminus__set__def,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A6: set_o] :
          ( collect_o
          @ ( uminus_uminus_o_o
            @ ^ [X3: $o] : ( member_o @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6922_uminus__set__def,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ( uminus6401447641752708672_nat_o
            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6923_uminus__set__def,axiom,
    ( uminus3098529973357106300at_rat
    = ( ^ [A6: set_set_nat_rat] :
          ( collect_set_nat_rat
          @ ( uminus6216118484121566985_rat_o
            @ ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6924_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X3: nat] : ( member_nat @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6925_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X3: int] : ( member_int @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6926_uminus__set__def,axiom,
    ( uminus6988975074191911878at_rat
    = ( ^ [A6: set_nat_rat] :
          ( collect_nat_rat
          @ ( uminus8974390361584276543_rat_o
            @ ^ [X3: nat > rat] : ( member_nat_rat @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6927_Collect__neg__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X3: set_nat] :
            ~ ( P @ X3 ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6928_Collect__neg__eq,axiom,
    ! [P: set_nat_rat > $o] :
      ( ( collect_set_nat_rat
        @ ^ [X3: set_nat_rat] :
            ~ ( P @ X3 ) )
      = ( uminus3098529973357106300at_rat @ ( collect_set_nat_rat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6929_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X3: nat] :
            ~ ( P @ X3 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6930_Collect__neg__eq,axiom,
    ! [P: int > $o] :
      ( ( collect_int
        @ ^ [X3: int] :
            ~ ( P @ X3 ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6931_Collect__neg__eq,axiom,
    ! [P: ( nat > rat ) > $o] :
      ( ( collect_nat_rat
        @ ^ [X3: nat > rat] :
            ~ ( P @ X3 ) )
      = ( uminus6988975074191911878at_rat @ ( collect_nat_rat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6932_Compl__eq,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A6: set_o] :
          ( collect_o
          @ ^ [X3: $o] :
              ~ ( member_o @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6933_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X3: set_nat] :
              ~ ( member_set_nat @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6934_Compl__eq,axiom,
    ( uminus3098529973357106300at_rat
    = ( ^ [A6: set_set_nat_rat] :
          ( collect_set_nat_rat
          @ ^ [X3: set_nat_rat] :
              ~ ( member_set_nat_rat @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6935_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ~ ( member_nat @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6936_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ^ [X3: int] :
              ~ ( member_int @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6937_Compl__eq,axiom,
    ( uminus6988975074191911878at_rat
    = ( ^ [A6: set_nat_rat] :
          ( collect_nat_rat
          @ ^ [X3: nat > rat] :
              ~ ( member_nat_rat @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_6938_not__finite__existsD,axiom,
    ! [P: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
     => ? [X_1: set_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6939_not__finite__existsD,axiom,
    ! [P: set_nat_rat > $o] :
      ( ~ ( finite6430367030675640852at_rat @ ( collect_set_nat_rat @ P ) )
     => ? [X_1: set_nat_rat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6940_not__finite__existsD,axiom,
    ! [P: ( nat > rat ) > $o] :
      ( ~ ( finite7830837933032798814at_rat @ ( collect_nat_rat @ P ) )
     => ? [X_1: nat > rat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6941_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6942_not__finite__existsD,axiom,
    ! [P: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P ) )
     => ? [X_1: int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6943_not__finite__existsD,axiom,
    ! [P: complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ? [X_1: complex] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6944_not__finite__existsD,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6945_not__finite__existsD,axiom,
    ! [P: extended_enat > $o] :
      ( ~ ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
     => ? [X_1: extended_enat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6946_pigeonhole__infinite__rel,axiom,
    ! [A2: set_o,B2: set_nat,R: $o > nat > $o] :
      ( ~ ( finite_finite_o @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B2 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A4: $o] :
                        ( ( member_o @ A4 @ A2 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6947_pigeonhole__infinite__rel,axiom,
    ! [A2: set_o,B2: set_int,R: $o > int > $o] :
      ( ~ ( finite_finite_o @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: int] :
              ( ( member_int @ X4 @ B2 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A4: $o] :
                        ( ( member_o @ A4 @ A2 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6948_pigeonhole__infinite__rel,axiom,
    ! [A2: set_o,B2: set_complex,R: $o > complex > $o] :
      ( ~ ( finite_finite_o @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: complex] :
              ( ( member_complex @ X4 @ B2 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A4: $o] :
                        ( ( member_o @ A4 @ A2 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6949_pigeonhole__infinite__rel,axiom,
    ! [A2: set_o,B2: set_Extended_enat,R: $o > extended_enat > $o] :
      ( ~ ( finite_finite_o @ A2 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ A2 )
             => ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ B2 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A4: $o] :
                        ( ( member_o @ A4 @ A2 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6950_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6951_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_int,R: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: int] :
              ( ( member_int @ X4 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6952_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_complex,R: nat > complex > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: complex] :
              ( ( member_complex @ X4 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6953_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_Extended_enat,R: nat > extended_enat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6954_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_nat,R: int > nat > $o] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B2 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A4: int] :
                        ( ( member_int @ A4 @ A2 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6955_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_int,R: int > int > $o] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: int] :
              ( ( member_int @ X4 @ B2 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A4: int] :
                        ( ( member_int @ A4 @ A2 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6956_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_6957_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_6958_minus__set__def,axiom,
    ( minus_minus_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ( minus_minus_o_o
            @ ^ [X3: $o] : ( member_o @ X3 @ A6 )
            @ ^ [X3: $o] : ( member_o @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6959_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6960_minus__set__def,axiom,
    ( minus_1626877696091177228at_rat
    = ( ^ [A6: set_set_nat_rat,B6: set_set_nat_rat] :
          ( collect_set_nat_rat
          @ ( minus_7664381017404958329_rat_o
            @ ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ A6 )
            @ ^ [X3: set_nat_rat] : ( member_set_nat_rat @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6961_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X3: int] : ( member_int @ X3 @ A6 )
            @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6962_minus__set__def,axiom,
    ( minus_1741603841019369558at_rat
    = ( ^ [A6: set_nat_rat,B6: set_nat_rat] :
          ( collect_nat_rat
          @ ( minus_8641456556474268591_rat_o
            @ ^ [X3: nat > rat] : ( member_nat_rat @ X3 @ A6 )
            @ ^ [X3: nat > rat] : ( member_nat_rat @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6963_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
            @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_6964_set__diff__eq,axiom,
    ( minus_minus_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ A6 )
              & ~ ( member_o @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6965_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X3: set_nat] :
              ( ( member_set_nat @ X3 @ A6 )
              & ~ ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6966_set__diff__eq,axiom,
    ( minus_1626877696091177228at_rat
    = ( ^ [A6: set_set_nat_rat,B6: set_set_nat_rat] :
          ( collect_set_nat_rat
          @ ^ [X3: set_nat_rat] :
              ( ( member_set_nat_rat @ X3 @ A6 )
              & ~ ( member_set_nat_rat @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6967_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ A6 )
              & ~ ( member_int @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6968_set__diff__eq,axiom,
    ( minus_1741603841019369558at_rat
    = ( ^ [A6: set_nat_rat,B6: set_nat_rat] :
          ( collect_nat_rat
          @ ^ [X3: nat > rat] :
              ( ( member_nat_rat @ X3 @ A6 )
              & ~ ( member_nat_rat @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6969_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ A6 )
              & ~ ( member_nat @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6970_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_6971_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z2: complex] :
                  ( ( power_power_complex @ Z2 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_6972_lambda__zero,axiom,
    ( ( ^ [H3: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_6973_lambda__zero,axiom,
    ( ( ^ [H3: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_6974_lambda__zero,axiom,
    ( ( ^ [H3: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_6975_lambda__zero,axiom,
    ( ( ^ [H3: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_6976_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_6977_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_code(2)
thf(fact_6978_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_code(2)
thf(fact_6979_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_code(2)
thf(fact_6980_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_code(2)
thf(fact_6981_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).

% numeral_code(2)
thf(fact_6982_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_code(2)
thf(fact_6983_finite__int__segment,axiom,
    ! [A: rat,B: rat] :
      ( finite_finite_rat
      @ ( collect_rat
        @ ^ [X3: rat] :
            ( ( member_rat @ X3 @ ring_1_Ints_rat )
            & ( ord_less_eq_rat @ A @ X3 )
            & ( ord_less_eq_rat @ X3 @ B ) ) ) ) ).

% finite_int_segment
thf(fact_6984_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_less_as_int
thf(fact_6985_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_leq_as_int
thf(fact_6986_finite__abs__int__segment,axiom,
    ! [A: real] :
      ( finite_finite_real
      @ ( collect_real
        @ ^ [K3: real] :
            ( ( member_real @ K3 @ ring_1_Ints_real )
            & ( ord_less_eq_real @ ( abs_abs_real @ K3 ) @ A ) ) ) ) ).

% finite_abs_int_segment
thf(fact_6987_finite__abs__int__segment,axiom,
    ! [A: rat] :
      ( finite_finite_rat
      @ ( collect_rat
        @ ^ [K3: rat] :
            ( ( member_rat @ K3 @ ring_1_Ints_rat )
            & ( ord_less_eq_rat @ ( abs_abs_rat @ K3 ) @ A ) ) ) ) ).

% finite_abs_int_segment
thf(fact_6988_card__less__Suc2,axiom,
    ! [M5: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M5 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M5 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M5 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_6989_card__less__Suc,axiom,
    ! [M5: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M5 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M5 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M5 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_6990_card__less,axiom,
    ! [M5: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M5 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M5 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_6991_list__update__code_I2_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( ( list_update_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_int @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_6992_list__update__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_nat @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_6993_list__update__code_I2_J,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_VEBT_VEBT @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_6994_set__update__subsetI,axiom,
    ! [Xs: list_o,A2: set_o,X: $o,I: nat] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_6995_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I: nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A2 )
     => ( ( member_set_nat @ X @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_6996_set__update__subsetI,axiom,
    ! [Xs: list_set_nat_rat,A2: set_set_nat_rat,X: set_nat_rat,I: nat] :
      ( ( ord_le4375437777232675859at_rat @ ( set_set_nat_rat2 @ Xs ) @ A2 )
     => ( ( member_set_nat_rat @ X @ A2 )
       => ( ord_le4375437777232675859at_rat @ ( set_set_nat_rat2 @ ( list_u886106648575569423at_rat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_6997_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_6998_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_6999_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_7000_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z2: real] :
              ( ( power_power_real @ Z2 @ N )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_7001_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_7002_card__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ord_less_eq_nat
        @ ( finite_card_real
          @ ( collect_real
            @ ^ [Z2: real] :
                ( ( power_power_real @ Z2 @ N )
                = one_one_real ) ) )
        @ N ) ) ).

% card_roots_unity
thf(fact_7003_card__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ord_less_eq_nat
        @ ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        @ N ) ) ).

% card_roots_unity
thf(fact_7004_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
              & ( ( size_s3451745648224563538omplex @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7005_finite__lists__length__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs2: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
              & ( ( size_s5460976970255530739at_nat @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7006_finite__lists__length__eq,axiom,
    ! [A2: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs2: list_Extended_enat] :
              ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A2 )
              & ( ( size_s3941691890525107288d_enat @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7007_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7008_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_nat @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7009_finite__lists__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_int @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7010_card__lists__length__eq,axiom,
    ! [A2: set_list_nat,N: nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite7325466520557071688st_nat
          @ ( collec5989764272469232197st_nat
            @ ^ [Xs2: list_list_nat] :
                ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A2 )
                & ( ( size_s3023201423986296836st_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_list_nat @ A2 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7011_card__lists__length__eq,axiom,
    ! [A2: set_set_nat,N: nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite5631907774883551598et_nat
          @ ( collect_list_set_nat
            @ ^ [Xs2: list_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A2 )
                & ( ( size_s3254054031482475050et_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_set_nat @ A2 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7012_card__lists__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite5120063068150530198omplex
          @ ( collect_list_complex
            @ ^ [Xs2: list_complex] :
                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
                & ( ( size_s3451745648224563538omplex @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_complex @ A2 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7013_card__lists__length__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite249151656366948015at_nat
          @ ( collec3343600615725829874at_nat
            @ ^ [Xs2: list_P6011104703257516679at_nat] :
                ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
                & ( ( size_s5460976970255530739at_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite711546835091564841at_nat @ A2 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7014_card__lists__length__eq,axiom,
    ! [A2: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite7441382602597825044d_enat
          @ ( collec8433460942617342167d_enat
            @ ^ [Xs2: list_Extended_enat] :
                ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A2 )
                & ( ( size_s3941691890525107288d_enat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite121521170596916366d_enat @ A2 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7015_card__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite5915292604075114978T_VEBT
          @ ( collec5608196760682091941T_VEBT
            @ ^ [Xs2: list_VEBT_VEBT] :
                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
                & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite7802652506058667612T_VEBT @ A2 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7016_card__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [Xs2: list_nat] :
                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
                & ( ( size_size_list_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_nat @ A2 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7017_card__lists__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_card_list_int
          @ ( collect_list_int
            @ ^ [Xs2: list_int] :
                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
                & ( ( size_size_list_int @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_int @ A2 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7018_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_7019_finite__lists__length__le,axiom,
    ! [A2: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7020_finite__lists__length__le,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs2: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7021_finite__lists__length__le,axiom,
    ! [A2: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs2: list_Extended_enat] :
              ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3941691890525107288d_enat @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7022_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7023_finite__lists__length__le,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7024_finite__lists__length__le,axiom,
    ! [A2: set_int,N: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7025_set__update__subset__insert,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I: nat,X: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X ) ) @ ( insert8211810215607154385at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_7026_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_7027_set__update__subset__insert,axiom,
    ! [Xs: list_o,I: nat,X: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X ) ) @ ( insert_o @ X @ ( set_o2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_7028_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_7029_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_7030_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_7031_set__update__memI,axiom,
    ! [N: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_7032_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X: set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_7033_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat_rat,X: set_nat_rat] :
      ( ( ord_less_nat @ N @ ( size_s3959913991096427681at_rat @ Xs ) )
     => ( member_set_nat_rat @ X @ ( set_set_nat_rat2 @ ( list_u886106648575569423at_rat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_7034_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_7035_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_7036_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_7037_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( list_update_int @ Xs @ I @ X )
          = Xs )
        = ( ( nth_int @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_7038_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
          = Xs )
        = ( ( nth_VEBT_VEBT @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_7039_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_7040_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
            = ( nth_int @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_7041_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
            = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_7042_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_7043_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_7044_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_7045_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ ( ord_less_nat @ X @ Mi )
            & ( ~ ( ord_less_nat @ X @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X )
                & ( ~ ( ord_less_nat @ Ma @ X )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_7046_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
      = ( ( X = Mi )
        | ( X = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_7047_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_7048_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A5 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B5 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_7049_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A5 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B5 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_7050_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_7051_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_7052_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A5 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B5 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_7053_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_7054_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                 => ( Y
                    = ( ~ ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_7055_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B5 ) ) ) )
       => ( ! [A5: $o] :
              ( ? [B5: $o] :
                  ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A5 @ $false ) ) ) )
         => ( ! [A5: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ( ? [N2: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ N2 ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A5 @ B5 ) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                   => ( Y
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                     => ( Y
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( Y
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                      @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_7056_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                      = ( ~ ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_7057_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A5 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B5 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_7058_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_7059_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_7060_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A5: $o] :
              ( ? [Uw2: $o] :
                  ( X
                  = ( vEBT_Leaf @ A5 @ Uw2 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A5
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A5
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A5: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ( ? [Va: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va ) ) )
                 => ~ ( ( B5
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B5
                       => ( ( A5
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A5
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_7061_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ! [Uu2: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu2 @ B5 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B5
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B5
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N2: nat] :
                  ( Xa2
                  = ( suc @ N2 ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_7062_of__int__code__if,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7063_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] :
          ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7064_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] :
          ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7065_of__int__code__if,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] :
          ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
          @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_rat
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7066_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] :
          ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7067_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_7068_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_7069_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B5 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A5 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( Xa2 = Mi2 )
                              | ( Xa2 = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_7070_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ B5 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( B5
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B5
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N2: nat] :
                    ( ( Xa2
                      = ( suc @ N2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_7071_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ Uw2 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A5
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A5
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A5: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A5 @ B5 ) )
                 => ! [Va: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( ( B5
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B5
                           => ( ( A5
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A5
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_7072_max__bot2,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ X @ bot_bot_set_real )
      = X ) ).

% max_bot2
thf(fact_7073_max__bot2,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ X @ bot_bot_set_o )
      = X ) ).

% max_bot2
thf(fact_7074_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_7075_max__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ X @ bot_bot_set_int )
      = X ) ).

% max_bot2
thf(fact_7076_max__bot2,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ X @ bot_bot_nat )
      = X ) ).

% max_bot2
thf(fact_7077_max__bot,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ bot_bot_set_real @ X )
      = X ) ).

% max_bot
thf(fact_7078_max__bot,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ bot_bot_set_o @ X )
      = X ) ).

% max_bot
thf(fact_7079_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_7080_max__bot,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X )
      = X ) ).

% max_bot
thf(fact_7081_max__bot,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X )
      = X ) ).

% max_bot
thf(fact_7082_max__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M2 @ N ) ) ) ).

% max_Suc_Suc
thf(fact_7083_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_7084_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_7085_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_7086_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_7087_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_7088_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_7089_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_7090_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ V ) ) )
      & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_7091_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_7092_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_7093_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_7094_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_7095_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_7096_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(3)
thf(fact_7097_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_7098_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_7099_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(3)
thf(fact_7100_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(3)
thf(fact_7101_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_7102_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(4)
thf(fact_7103_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_7104_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_7105_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(4)
thf(fact_7106_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ zero_z3403309356797280102nteger )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(4)
thf(fact_7107_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_7108_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_7109_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_7110_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_7111_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_7112_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_7113_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_7114_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_7115_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_7116_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(5)
thf(fact_7117_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_7118_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_7119_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(5)
thf(fact_7120_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_7121_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_7122_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(6)
thf(fact_7123_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_7124_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_7125_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(6)
thf(fact_7126_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_7127_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_7128_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_7129_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_7130_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_7131_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_7132_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_7133_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_7134_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_7135_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_7136_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_7137_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_7138_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_7139_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_7140_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_7141_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).

% of_nat_max
thf(fact_7142_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y ) ) ) ).

% of_nat_max
thf(fact_7143_max__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_max_set_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7144_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_max_rat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7145_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_max_num @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7146_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7147_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_max_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7148_max__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_max_set_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_7149_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_max_rat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_7150_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_max_num @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_7151_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_7152_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_max_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_7153_max__def,axiom,
    ( ord_max_set_int
    = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_7154_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_7155_max__def,axiom,
    ( ord_max_num
    = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_7156_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_7157_max__def,axiom,
    ( ord_max_int
    = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_7158_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_7159_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_7160_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_7161_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_7162_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_7163_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_7164_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_7165_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_7166_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_7167_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_7168_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_7169_nat__add__max__right,axiom,
    ! [M2: nat,N: nat,Q4: nat] :
      ( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N @ Q4 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ N ) @ ( plus_plus_nat @ M2 @ Q4 ) ) ) ).

% nat_add_max_right
thf(fact_7170_nat__add__max__left,axiom,
    ! [M2: nat,N: nat,Q4: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N ) @ Q4 )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q4 ) @ ( plus_plus_nat @ N @ Q4 ) ) ) ).

% nat_add_max_left
thf(fact_7171_nat__mult__max__left,axiom,
    ! [M2: nat,N: nat,Q4: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M2 @ N ) @ Q4 )
      = ( ord_max_nat @ ( times_times_nat @ M2 @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).

% nat_mult_max_left
thf(fact_7172_nat__mult__max__right,axiom,
    ! [M2: nat,N: nat,Q4: nat] :
      ( ( times_times_nat @ M2 @ ( ord_max_nat @ N @ Q4 ) )
      = ( ord_max_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q4 ) ) ) ).

% nat_mult_max_right
thf(fact_7173_max__def__raw,axiom,
    ( ord_max_set_int
    = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_7174_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_7175_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_7176_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_7177_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_7178_nat__minus__add__max,axiom,
    ! [N: nat,M2: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M2 ) @ M2 )
      = ( ord_max_nat @ N @ M2 ) ) ).

% nat_minus_add_max
thf(fact_7179_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_7180_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y
                    = ( vEBT_Leaf @ $false @ B5 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A5: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( vEBT_Leaf @ A5 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A5: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A5 @ B5 ) )
                 => ! [N2: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A5 @ B5 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                     => ( ( Y
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                       => ( ( Y
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( Y
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_7181_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B5 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A5 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( Xa2 = Mi2 )
                                  | ( Xa2 = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_7182_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_7183_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A5 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B5 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_7184_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A5 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B5 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_7185_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_7186_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_7187_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_7188_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                   => ( ( Y
                        = ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_7189_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_7190_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_7191_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7192_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7193_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7194_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7195_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7196_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7197_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7198_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7199_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7200_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7201_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ( ord_less_real @ X @ Z )
        & ( ord_less_real @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7202_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ( ord_less_rat @ X @ Z )
        & ( ord_less_rat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7203_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
      = ( ( ord_less_num @ X @ Z )
        & ( ord_less_num @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7204_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        & ( ord_less_nat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7205_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        & ( ord_less_int @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7206_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_7207_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_7208_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_7209_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_7210_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_7211_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_7212_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_7213_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_7214_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_7215_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_7216_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_7217_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_7218_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_7219_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_7220_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_7221_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_7222_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_7223_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_7224_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_7225_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_7226_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_7227_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_7228_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_7229_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_7230_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_7231_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_7232_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_7233_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_7234_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_7235_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_7236_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_7237_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_7238_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( A4
          = ( ord_max_rat @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_7239_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( A4
          = ( ord_max_num @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_7240_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( A4
          = ( ord_max_nat @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_7241_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( A4
          = ( ord_max_int @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_7242_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_7243_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_7244_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_7245_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_7246_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_7247_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_7248_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_7249_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_7250_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z @ X )
        | ( ord_less_eq_rat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_7251_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_eq_num @ Z @ X )
        | ( ord_less_eq_num @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_7252_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z @ X )
        | ( ord_less_eq_nat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_7253_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z @ X )
        | ( ord_less_eq_int @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_7254_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_max_rat @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_7255_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_max_num @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_7256_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_max_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_7257_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_max_int @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_7258_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_max_rat @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_7259_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_max_num @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_7260_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_max_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_7261_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_max_int @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_7262_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_7263_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_7264_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_7265_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_7266_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_7267_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_7268_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_7269_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_7270_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_real @ Z @ X )
        | ( ord_less_real @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7271_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z @ X )
        | ( ord_less_rat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7272_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_num @ Z @ X )
        | ( ord_less_num @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7273_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        | ( ord_less_nat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7274_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        | ( ord_less_int @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7275_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7276_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_7277_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_7278_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_7279_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_7280_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( A4
            = ( ord_max_real @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7281_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( A4
            = ( ord_max_rat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7282_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( A4
            = ( ord_max_num @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7283_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( A4
            = ( ord_max_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7284_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( A4
            = ( ord_max_int @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7285_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7286_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_7287_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_7288_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_7289_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_7290_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ B )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7291_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_7292_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_7293_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_7294_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_7295_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7296_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] :
          ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L3: nat] : ( times_times_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ L3 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7297_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] :
          ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L3: nat] : ( times_times_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ L3 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7298_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] :
          ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L3: nat] : ( times_times_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ L3 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7299_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_7300_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_7301_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_7302_ln__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X )
          = ( suminf_real
            @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N4 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_7303_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N4: nat,A4: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7304_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,A4: int] : ( if_int @ ( N4 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7305_arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arctan @ X )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_7306_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_7307_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_7308_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_7309_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_7310_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7311_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7312_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6464643781859351333omplex @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6464643781859351333omplex @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7313_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7314_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7315_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7316_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7317_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_7318_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_7319_prod__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B: nat] :
      ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A4: nat] : ( times_times_complex @ ( F @ A4 ) )
        @ A
        @ B
        @ one_one_complex ) ) ).

% prod_atLeastAtMost_code
thf(fact_7320_prod__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A4: nat] : ( times_times_real @ ( F @ A4 ) )
        @ A
        @ B
        @ one_one_real ) ) ).

% prod_atLeastAtMost_code
thf(fact_7321_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A4: nat] : ( times_times_rat @ ( F @ A4 ) )
        @ A
        @ B
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_7322_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A4: nat] : ( times_times_nat @ ( F @ A4 ) )
        @ A
        @ B
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_7323_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A4: nat] : ( times_times_int @ ( F @ A4 ) )
        @ A
        @ B
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_7324_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: nat > nat,M2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_7325_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: nat > int,M2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_7326_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: nat > nat,M2: nat,K: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G2 @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_7327_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: nat > int,M2: nat,K: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G2 @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_7328_prod_OatLeastAtMost__rev,axiom,
    ! [G2: nat > nat,N: nat,M2: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_7329_prod_OatLeastAtMost__rev,axiom,
    ! [G2: nat > int,N: nat,M2: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_7330_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7331_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7332_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7333_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7334_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_real @ ( G2 @ M2 ) @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7335_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_rat @ ( G2 @ M2 ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7336_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_nat @ ( G2 @ M2 ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7337_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_int @ ( G2 @ M2 ) @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7338_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_real @ ( G2 @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7339_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G2 @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7340_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G2 @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7341_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_int @ ( G2 @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7342_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( times_times_real @ ( G2 @ M2 )
          @ ( groups129246275422532515t_real
            @ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7343_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G2 @ M2 )
          @ ( groups73079841787564623at_rat
            @ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7344_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G2 @ M2 )
          @ ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7345_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( times_times_int @ ( G2 @ M2 )
          @ ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7346_fact__prod,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N4: nat] :
          ( semiri1314217659103216013at_int
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).

% fact_prod
thf(fact_7347_fact__prod,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N4: nat] :
          ( semiri681578069525770553at_rat
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).

% fact_prod
thf(fact_7348_fact__prod,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N4: nat] :
          ( semiri1316708129612266289at_nat
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).

% fact_prod
thf(fact_7349_fact__prod,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N4: nat] :
          ( semiri5074537144036343181t_real
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).

% fact_prod
thf(fact_7350_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > real,P6: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P6 ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7351_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > rat,P6: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P6 ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7352_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > nat,P6: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P6 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7353_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > int,P6: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P6 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7354_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb2 @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb2 @ Xa2 )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_7355_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F5: nat > nat > nat,A4: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A4 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F5 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F5 @ A4 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_7356_fact__eq__fact__times,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri1408675320244567234ct_nat @ M2 )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M2 ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_7357_monoseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).

% monoseq_realpow
thf(fact_7358_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( groups129246275422532515t_real
        @ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7359_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7360_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7361_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7362_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_7363_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_7364_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_7365_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_7366_pochhammer__prod__rev,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A4: real,N4: nat] :
          ( groups129246275422532515t_real
          @ ^ [I4: nat] : ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N4 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7367_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A4: rat,N4: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N4 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7368_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A4: nat,N4: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N4 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7369_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A4: int,N4: nat] :
          ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N4 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7370_fact__div__fact,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X3: nat] : X3
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) ) ).

% fact_div_fact
thf(fact_7371_prod_Oin__pairs,axiom,
    ! [G2: nat > real,M2: nat,N: nat] :
      ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I4: nat] : ( times_times_real @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_7372_prod_Oin__pairs,axiom,
    ! [G2: nat > rat,M2: nat,N: nat] :
      ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( times_times_rat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_7373_prod_Oin__pairs,axiom,
    ! [G2: nat > nat,M2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( times_times_nat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_7374_prod_Oin__pairs,axiom,
    ! [G2: nat > int,M2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( times_times_int @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_7375_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( groups129246275422532515t_real
        @ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7376_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7377_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7378_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7379_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_7380_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_7381_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_7382_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( divide_divide_rat
        @ ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7383_gbinomial__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( divide_divide_real
        @ ( groups129246275422532515t_real
          @ ^ [I4: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7384_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7385_gbinomial__Suc,axiom,
    ! [A: int,K: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7386_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_7387_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N4: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7388_fact__code,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N4: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7389_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N4: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7390_fact__code,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N4: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7391_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A4: complex,N4: nat] :
          ( if_complex @ ( N4 = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_7392_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A4: int,N4: nat] :
          ( if_int @ ( N4 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_7393_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A4: real,N4: nat] :
          ( if_real @ ( N4 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_7394_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A4: rat,N4: nat] :
          ( if_rat @ ( N4 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_7395_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A4: nat,N4: nat] :
          ( if_nat @ ( N4 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_7396_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G2: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1681761925125756287l_real @ G2 @ ( insert_real @ X @ A2 ) )
          = ( times_times_real @ ( G2 @ X ) @ ( groups1681761925125756287l_real @ G2 @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_7397_prod_Oinsert,axiom,
    ! [A2: set_o,X: $o,G2: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( groups234877984723959775o_real @ G2 @ ( insert_o @ X @ A2 ) )
          = ( times_times_real @ ( G2 @ X ) @ ( groups234877984723959775o_real @ G2 @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_7398_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G2: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups129246275422532515t_real @ G2 @ ( insert_nat @ X @ A2 ) )
          = ( times_times_real @ ( G2 @ X ) @ ( groups129246275422532515t_real @ G2 @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_7399_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G2: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups2316167850115554303t_real @ G2 @ ( insert_int @ X @ A2 ) )
          = ( times_times_real @ ( G2 @ X ) @ ( groups2316167850115554303t_real @ G2 @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_7400_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups766887009212190081x_real @ G2 @ ( insert_complex @ X @ A2 ) )
          = ( times_times_real @ ( G2 @ X ) @ ( groups766887009212190081x_real @ G2 @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_7401_prod_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ~ ( member_Extended_enat @ X @ A2 )
       => ( ( groups97031904164794029t_real @ G2 @ ( insert_Extended_enat @ X @ A2 ) )
          = ( times_times_real @ ( G2 @ X ) @ ( groups97031904164794029t_real @ G2 @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_7402_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G2: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups4061424788464935467al_rat @ G2 @ ( insert_real @ X @ A2 ) )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups4061424788464935467al_rat @ G2 @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_7403_prod_Oinsert,axiom,
    ! [A2: set_o,X: $o,G2: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( groups2869687844427037835_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups2869687844427037835_o_rat @ G2 @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_7404_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups73079841787564623at_rat @ G2 @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_7405_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G2: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups1072433553688619179nt_rat @ G2 @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_7406_prod_Odelta,axiom,
    ! [S2: set_o,A: $o,B: $o > complex] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups4859619685533338977omplex
              @ ^ [K3: $o] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups4859619685533338977omplex
              @ ^ [K3: $o] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7407_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7408_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7409_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7410_prod_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups4622424608036095791omplex
              @ ^ [K3: extended_enat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups4622424608036095791omplex
              @ ^ [K3: extended_enat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7411_prod_Odelta,axiom,
    ! [S2: set_o,A: $o,B: $o > real] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups234877984723959775o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups234877984723959775o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7412_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7413_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7414_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7415_prod_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7416_prod_Odelta_H,axiom,
    ! [S2: set_o,A: $o,B: $o > complex] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups4859619685533338977omplex
              @ ^ [K3: $o] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups4859619685533338977omplex
              @ ^ [K3: $o] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7417_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7418_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7419_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7420_prod_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups4622424608036095791omplex
              @ ^ [K3: extended_enat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups4622424608036095791omplex
              @ ^ [K3: extended_enat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7421_prod_Odelta_H,axiom,
    ! [S2: set_o,A: $o,B: $o > real] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups234877984723959775o_real
              @ ^ [K3: $o] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups234877984723959775o_real
              @ ^ [K3: $o] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7422_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7423_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7424_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7425_prod_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S2 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7426_suminf__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( ( suminf_real @ ( power_power_real @ C ) )
        = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% suminf_geometric
thf(fact_7427_suminf__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( ( suminf_complex @ ( power_power_complex @ C ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% suminf_geometric
thf(fact_7428_prod_Oinfinite,axiom,
    ! [A2: set_nat,G2: nat > complex] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups6464643781859351333omplex @ G2 @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7429_prod_Oinfinite,axiom,
    ! [A2: set_int,G2: int > complex] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups7440179247065528705omplex @ G2 @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7430_prod_Oinfinite,axiom,
    ! [A2: set_complex,G2: complex > complex] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G2 @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7431_prod_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > complex] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4622424608036095791omplex @ G2 @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7432_prod_Oinfinite,axiom,
    ! [A2: set_nat,G2: nat > real] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G2 @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7433_prod_Oinfinite,axiom,
    ! [A2: set_int,G2: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G2 @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7434_prod_Oinfinite,axiom,
    ! [A2: set_complex,G2: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G2 @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7435_prod_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > real] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups97031904164794029t_real @ G2 @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7436_prod_Oinfinite,axiom,
    ! [A2: set_nat,G2: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups73079841787564623at_rat @ G2 @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_7437_prod_Oinfinite,axiom,
    ! [A2: set_int,G2: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G2 @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_7438_prod_Oempty,axiom,
    ! [G2: real > complex] :
      ( ( groups713298508707869441omplex @ G2 @ bot_bot_set_real )
      = one_one_complex ) ).

% prod.empty
thf(fact_7439_prod_Oempty,axiom,
    ! [G2: real > real] :
      ( ( groups1681761925125756287l_real @ G2 @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_7440_prod_Oempty,axiom,
    ! [G2: real > rat] :
      ( ( groups4061424788464935467al_rat @ G2 @ bot_bot_set_real )
      = one_one_rat ) ).

% prod.empty
thf(fact_7441_prod_Oempty,axiom,
    ! [G2: real > nat] :
      ( ( groups4696554848551431203al_nat @ G2 @ bot_bot_set_real )
      = one_one_nat ) ).

% prod.empty
thf(fact_7442_prod_Oempty,axiom,
    ! [G2: real > int] :
      ( ( groups4694064378042380927al_int @ G2 @ bot_bot_set_real )
      = one_one_int ) ).

% prod.empty
thf(fact_7443_prod_Oempty,axiom,
    ! [G2: $o > complex] :
      ( ( groups4859619685533338977omplex @ G2 @ bot_bot_set_o )
      = one_one_complex ) ).

% prod.empty
thf(fact_7444_prod_Oempty,axiom,
    ! [G2: $o > real] :
      ( ( groups234877984723959775o_real @ G2 @ bot_bot_set_o )
      = one_one_real ) ).

% prod.empty
thf(fact_7445_prod_Oempty,axiom,
    ! [G2: $o > rat] :
      ( ( groups2869687844427037835_o_rat @ G2 @ bot_bot_set_o )
      = one_one_rat ) ).

% prod.empty
thf(fact_7446_prod_Oempty,axiom,
    ! [G2: $o > nat] :
      ( ( groups3504817904513533571_o_nat @ G2 @ bot_bot_set_o )
      = one_one_nat ) ).

% prod.empty
thf(fact_7447_prod_Oempty,axiom,
    ! [G2: $o > int] :
      ( ( groups3502327434004483295_o_int @ G2 @ bot_bot_set_o )
      = one_one_int ) ).

% prod.empty
thf(fact_7448_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu3: nat] : one_one_nat
        @ A2 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_7449_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu3: nat] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_7450_prod_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu3: int] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_7451_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N4: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_7452_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N4: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_7453_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N4: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_7454_prod__zero__iff,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups129246275422532515t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
              & ( ( F @ X3 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7455_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups2316167850115554303t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A2 )
              & ( ( F @ X3 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7456_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups766887009212190081x_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ( F @ X3 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7457_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups97031904164794029t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A2 )
              & ( ( F @ X3 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7458_prod__zero__iff,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups73079841787564623at_rat @ F @ A2 )
          = zero_zero_rat )
        = ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
              & ( ( F @ X3 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7459_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups1072433553688619179nt_rat @ F @ A2 )
          = zero_zero_rat )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A2 )
              & ( ( F @ X3 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7460_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups225925009352817453ex_rat @ F @ A2 )
          = zero_zero_rat )
        = ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ( F @ X3 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7461_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2245840878043517529at_rat @ F @ A2 )
          = zero_zero_rat )
        = ( ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A2 )
              & ( ( F @ X3 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7462_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
          = zero_zero_nat )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A2 )
              & ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7463_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups861055069439313189ex_nat @ F @ A2 )
          = zero_zero_nat )
        = ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7464_prod__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7465_prod__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups861055069439313189ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7466_prod__eq__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ( groups4077766827762148844at_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A2 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7467_prod__eq__1__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2880970938130013265at_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A2 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7468_prod__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups708209901874060359at_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7469_prod__pos__nat__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
        = ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7470_prod__pos__nat__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) )
        = ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7471_prod__pos__nat__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups4077766827762148844at_nat @ F @ A2 ) )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7472_prod__pos__nat__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) )
        = ( ! [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7473_prod__pos__nat__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7474_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : X3
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_7475_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : X3
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_7476_prod_Oneutral,axiom,
    ! [A2: set_nat,G2: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( G2 @ X4 )
            = one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G2 @ A2 )
        = one_one_nat ) ) ).

% prod.neutral
thf(fact_7477_prod_Oneutral,axiom,
    ! [A2: set_nat,G2: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( G2 @ X4 )
            = one_one_int ) )
     => ( ( groups705719431365010083at_int @ G2 @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_7478_prod_Oneutral,axiom,
    ! [A2: set_int,G2: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ( G2 @ X4 )
            = one_one_int ) )
     => ( ( groups1705073143266064639nt_int @ G2 @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_7479_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: $o > complex,A2: set_o] :
      ( ( ( groups4859619685533338977omplex @ G2 @ A2 )
       != one_one_complex )
     => ~ ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7480_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: nat > complex,A2: set_nat] :
      ( ( ( groups6464643781859351333omplex @ G2 @ A2 )
       != one_one_complex )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7481_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: int > complex,A2: set_int] :
      ( ( ( groups7440179247065528705omplex @ G2 @ A2 )
       != one_one_complex )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7482_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: $o > real,A2: set_o] :
      ( ( ( groups234877984723959775o_real @ G2 @ A2 )
       != one_one_real )
     => ~ ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7483_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: nat > real,A2: set_nat] :
      ( ( ( groups129246275422532515t_real @ G2 @ A2 )
       != one_one_real )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7484_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: int > real,A2: set_int] :
      ( ( ( groups2316167850115554303t_real @ G2 @ A2 )
       != one_one_real )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7485_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: $o > rat,A2: set_o] :
      ( ( ( groups2869687844427037835_o_rat @ G2 @ A2 )
       != one_one_rat )
     => ~ ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7486_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: nat > rat,A2: set_nat] :
      ( ( ( groups73079841787564623at_rat @ G2 @ A2 )
       != one_one_rat )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7487_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: int > rat,A2: set_int] :
      ( ( ( groups1072433553688619179nt_rat @ G2 @ A2 )
       != one_one_rat )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7488_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: $o > nat,A2: set_o] :
      ( ( ( groups3504817904513533571_o_nat @ G2 @ A2 )
       != one_one_nat )
     => ~ ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7489_prod_Oswap__restrict,axiom,
    ! [A2: set_o,B2: set_nat,G2: $o > nat > nat,R: $o > nat > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups3504817904513533571_o_nat
            @ ^ [X3: $o] :
                ( groups708209901874060359at_nat @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y2: nat] :
                ( groups3504817904513533571_o_nat
                @ ^ [X3: $o] : ( G2 @ X3 @ Y2 )
                @ ( collect_o
                  @ ^ [X3: $o] :
                      ( ( member_o @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% prod.swap_restrict
thf(fact_7490_prod_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G2: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups1707563613775114915nt_nat
            @ ^ [X3: int] :
                ( groups708209901874060359at_nat @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y2: nat] :
                ( groups1707563613775114915nt_nat
                @ ^ [X3: int] : ( G2 @ X3 @ Y2 )
                @ ( collect_int
                  @ ^ [X3: int] :
                      ( ( member_int @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% prod.swap_restrict
thf(fact_7491_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G2: complex > nat > nat,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups861055069439313189ex_nat
            @ ^ [X3: complex] :
                ( groups708209901874060359at_nat @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y2: nat] :
                ( groups861055069439313189ex_nat
                @ ^ [X3: complex] : ( G2 @ X3 @ Y2 )
                @ ( collect_complex
                  @ ^ [X3: complex] :
                      ( ( member_complex @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% prod.swap_restrict
thf(fact_7492_prod_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B2: set_nat,G2: extended_enat > nat > nat,R: extended_enat > nat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups2880970938130013265at_nat
            @ ^ [X3: extended_enat] :
                ( groups708209901874060359at_nat @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y2: nat] :
                ( groups2880970938130013265at_nat
                @ ^ [X3: extended_enat] : ( G2 @ X3 @ Y2 )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X3: extended_enat] :
                      ( ( member_Extended_enat @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% prod.swap_restrict
thf(fact_7493_prod_Oswap__restrict,axiom,
    ! [A2: set_o,B2: set_nat,G2: $o > nat > int,R: $o > nat > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups3502327434004483295_o_int
            @ ^ [X3: $o] :
                ( groups705719431365010083at_int @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y2: nat] :
                ( groups3502327434004483295_o_int
                @ ^ [X3: $o] : ( G2 @ X3 @ Y2 )
                @ ( collect_o
                  @ ^ [X3: $o] :
                      ( ( member_o @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% prod.swap_restrict
thf(fact_7494_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G2: complex > nat > int,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups858564598930262913ex_int
            @ ^ [X3: complex] :
                ( groups705719431365010083at_int @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y2: nat] :
                ( groups858564598930262913ex_int
                @ ^ [X3: complex] : ( G2 @ X3 @ Y2 )
                @ ( collect_complex
                  @ ^ [X3: complex] :
                      ( ( member_complex @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% prod.swap_restrict
thf(fact_7495_prod_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B2: set_nat,G2: extended_enat > nat > int,R: extended_enat > nat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups2878480467620962989at_int
            @ ^ [X3: extended_enat] :
                ( groups705719431365010083at_int @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y2: nat] :
                ( groups2878480467620962989at_int
                @ ^ [X3: extended_enat] : ( G2 @ X3 @ Y2 )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X3: extended_enat] :
                      ( ( member_Extended_enat @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% prod.swap_restrict
thf(fact_7496_prod_Oswap__restrict,axiom,
    ! [A2: set_o,B2: set_int,G2: $o > int > int,R: $o > int > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups3502327434004483295_o_int
            @ ^ [X3: $o] :
                ( groups1705073143266064639nt_int @ ( G2 @ X3 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y2: int] :
                ( groups3502327434004483295_o_int
                @ ^ [X3: $o] : ( G2 @ X3 @ Y2 )
                @ ( collect_o
                  @ ^ [X3: $o] :
                      ( ( member_o @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% prod.swap_restrict
thf(fact_7497_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_int,G2: complex > int > int,R: complex > int > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups858564598930262913ex_int
            @ ^ [X3: complex] :
                ( groups1705073143266064639nt_int @ ( G2 @ X3 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y2: int] :
                ( groups858564598930262913ex_int
                @ ^ [X3: complex] : ( G2 @ X3 @ Y2 )
                @ ( collect_complex
                  @ ^ [X3: complex] :
                      ( ( member_complex @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% prod.swap_restrict
thf(fact_7498_prod_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B2: set_int,G2: extended_enat > int > int,R: extended_enat > int > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups2878480467620962989at_int
            @ ^ [X3: extended_enat] :
                ( groups1705073143266064639nt_int @ ( G2 @ X3 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y2: int] :
                ( groups2878480467620962989at_int
                @ ^ [X3: extended_enat] : ( G2 @ X3 @ Y2 )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X3: extended_enat] :
                      ( ( member_Extended_enat @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% prod.swap_restrict
thf(fact_7499_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_7500_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_7501_prod__nonneg,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_7502_prod__mono,axiom,
    ! [A2: set_o,F: $o > real,G2: $o > real] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups234877984723959775o_real @ F @ A2 ) @ ( groups234877984723959775o_real @ G2 @ A2 ) ) ) ).

% prod_mono
thf(fact_7503_prod__mono,axiom,
    ! [A2: set_nat,F: nat > real,G2: nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G2 @ A2 ) ) ) ).

% prod_mono
thf(fact_7504_prod__mono,axiom,
    ! [A2: set_int,F: int > real,G2: int > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G2 @ A2 ) ) ) ).

% prod_mono
thf(fact_7505_prod__mono,axiom,
    ! [A2: set_o,F: $o > rat,G2: $o > rat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) @ ( groups2869687844427037835_o_rat @ G2 @ A2 ) ) ) ).

% prod_mono
thf(fact_7506_prod__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G2: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G2 @ A2 ) ) ) ).

% prod_mono
thf(fact_7507_prod__mono,axiom,
    ! [A2: set_int,F: int > rat,G2: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G2 @ A2 ) ) ) ).

% prod_mono
thf(fact_7508_prod__mono,axiom,
    ! [A2: set_o,F: $o > nat,G2: $o > nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) @ ( groups3504817904513533571_o_nat @ G2 @ A2 ) ) ) ).

% prod_mono
thf(fact_7509_prod__mono,axiom,
    ! [A2: set_int,F: int > nat,G2: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G2 @ A2 ) ) ) ).

% prod_mono
thf(fact_7510_prod__mono,axiom,
    ! [A2: set_o,F: $o > int,G2: $o > int] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I2 ) )
            & ( ord_less_eq_int @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
     => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A2 ) @ ( groups3502327434004483295_o_int @ G2 @ A2 ) ) ) ).

% prod_mono
thf(fact_7511_prod__mono,axiom,
    ! [A2: set_nat,F: nat > nat,G2: nat > nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ G2 @ A2 ) ) ) ).

% prod_mono
thf(fact_7512_prod__pos,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_7513_prod__pos,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_7514_prod__pos,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_7515_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > real] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups234877984723959775o_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7516_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7517_prod__ge__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7518_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > rat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7519_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7520_prod__ge__1,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7521_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7522_prod__ge__1,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7523_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > int] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_int @ one_one_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7524_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7525_prod__zero,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_real ) )
       => ( ( groups129246275422532515t_real @ F @ A2 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_7526_prod__zero,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ? [X2: int] :
            ( ( member_int @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_real ) )
       => ( ( groups2316167850115554303t_real @ F @ A2 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_7527_prod__zero,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ? [X2: complex] :
            ( ( member_complex @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_real ) )
       => ( ( groups766887009212190081x_real @ F @ A2 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_7528_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ? [X2: extended_enat] :
            ( ( member_Extended_enat @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_real ) )
       => ( ( groups97031904164794029t_real @ F @ A2 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_7529_prod__zero,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_rat ) )
       => ( ( groups73079841787564623at_rat @ F @ A2 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_7530_prod__zero,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ? [X2: int] :
            ( ( member_int @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_rat ) )
       => ( ( groups1072433553688619179nt_rat @ F @ A2 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_7531_prod__zero,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ? [X2: complex] :
            ( ( member_complex @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_rat ) )
       => ( ( groups225925009352817453ex_rat @ F @ A2 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_7532_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ? [X2: extended_enat] :
            ( ( member_Extended_enat @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_rat ) )
       => ( ( groups2245840878043517529at_rat @ F @ A2 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_7533_prod__zero,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ? [X2: int] :
            ( ( member_int @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_nat ) )
       => ( ( groups1707563613775114915nt_nat @ F @ A2 )
          = zero_zero_nat ) ) ) ).

% prod_zero
thf(fact_7534_prod__zero,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ? [X2: complex] :
            ( ( member_complex @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_nat ) )
       => ( ( groups861055069439313189ex_nat @ F @ A2 )
          = zero_zero_nat ) ) ) ).

% prod_zero
thf(fact_7535_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > real,Y: $o > real] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7536_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7537_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7538_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7539_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: extended_enat > real,Y: extended_enat > real] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7540_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > rat,Y: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7541_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7542_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > rat,Y: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7543_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > rat,Y: complex > rat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7544_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: extended_enat > rat,Y: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7545_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > complex,Y: $o > complex] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7546_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7547_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7548_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7549_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: extended_enat > complex,Y: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7550_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > real,Y: $o > real] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7551_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7552_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7553_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7554_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: extended_enat > real,Y: extended_enat > real] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_real ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7555_prod_Ointer__filter,axiom,
    ! [A2: set_o,G2: $o > complex,P: $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups4859619685533338977omplex @ G2
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups4859619685533338977omplex
          @ ^ [X3: $o] : ( if_complex @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7556_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G2: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups6464643781859351333omplex @ G2
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups6464643781859351333omplex
          @ ^ [X3: nat] : ( if_complex @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7557_prod_Ointer__filter,axiom,
    ! [A2: set_int,G2: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups7440179247065528705omplex @ G2
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups7440179247065528705omplex
          @ ^ [X3: int] : ( if_complex @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7558_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G2: complex > complex,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G2
          @ ( collect_complex
            @ ^ [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups3708469109370488835omplex
          @ ^ [X3: complex] : ( if_complex @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7559_prod_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > complex,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4622424608036095791omplex @ G2
          @ ( collec4429806609662206161d_enat
            @ ^ [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups4622424608036095791omplex
          @ ^ [X3: extended_enat] : ( if_complex @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7560_prod_Ointer__filter,axiom,
    ! [A2: set_o,G2: $o > real,P: $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups234877984723959775o_real @ G2
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups234877984723959775o_real
          @ ^ [X3: $o] : ( if_real @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7561_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G2: nat > real,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G2
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups129246275422532515t_real
          @ ^ [X3: nat] : ( if_real @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7562_prod_Ointer__filter,axiom,
    ! [A2: set_int,G2: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G2
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups2316167850115554303t_real
          @ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7563_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G2: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G2
          @ ( collect_complex
            @ ^ [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups766887009212190081x_real
          @ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7564_prod_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > real,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups97031904164794029t_real @ G2
          @ ( collec4429806609662206161d_enat
            @ ^ [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups97031904164794029t_real
          @ ^ [X3: extended_enat] : ( if_real @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7565_prod__le__1,axiom,
    ! [A2: set_o,F: $o > real] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
            & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups234877984723959775o_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7566_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
            & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7567_prod__le__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
            & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7568_prod__le__1,axiom,
    ! [A2: set_o,F: $o > rat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
            & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7569_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
            & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7570_prod__le__1,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
            & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7571_prod__le__1,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
            & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_7572_prod__le__1,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
            & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_7573_prod__le__1,axiom,
    ! [A2: set_o,F: $o > int] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) )
            & ( ord_less_eq_int @ ( F @ X4 ) @ one_one_int ) ) )
     => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A2 ) @ one_one_int ) ) ).

% prod_le_1
thf(fact_7574_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
            & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_7575_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S2: set_nat,H: nat > complex,G2: nat > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X1: complex,Y1: complex,X24: complex,Y24: complex] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X24 @ Y24 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups6464643781859351333omplex @ H @ S2 ) @ ( groups6464643781859351333omplex @ G2 @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_7576_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S2: set_int,H: int > complex,G2: int > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X1: complex,Y1: complex,X24: complex,Y24: complex] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups7440179247065528705omplex @ H @ S2 ) @ ( groups7440179247065528705omplex @ G2 @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_7577_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S2: set_complex,H: complex > complex,G2: complex > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X1: complex,Y1: complex,X24: complex,Y24: complex] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups3708469109370488835omplex @ H @ S2 ) @ ( groups3708469109370488835omplex @ G2 @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_7578_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S2: set_Extended_enat,H: extended_enat > complex,G2: extended_enat > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X1: complex,Y1: complex,X24: complex,Y24: complex] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X24 @ Y24 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups4622424608036095791omplex @ H @ S2 ) @ ( groups4622424608036095791omplex @ G2 @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_7579_prod_Orelated,axiom,
    ! [R: real > real > $o,S2: set_nat,H: nat > real,G2: nat > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X24: real,Y24: real] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X24 @ Y24 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups129246275422532515t_real @ H @ S2 ) @ ( groups129246275422532515t_real @ G2 @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_7580_prod_Orelated,axiom,
    ! [R: real > real > $o,S2: set_int,H: int > real,G2: int > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X24: real,Y24: real] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups2316167850115554303t_real @ H @ S2 ) @ ( groups2316167850115554303t_real @ G2 @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_7581_prod_Orelated,axiom,
    ! [R: real > real > $o,S2: set_complex,H: complex > real,G2: complex > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X24: real,Y24: real] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups766887009212190081x_real @ H @ S2 ) @ ( groups766887009212190081x_real @ G2 @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_7582_prod_Orelated,axiom,
    ! [R: real > real > $o,S2: set_Extended_enat,H: extended_enat > real,G2: extended_enat > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X24: real,Y24: real] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X24 @ Y24 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups97031904164794029t_real @ H @ S2 ) @ ( groups97031904164794029t_real @ G2 @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_7583_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S2: set_nat,H: nat > rat,G2: nat > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X1: rat,Y1: rat,X24: rat,Y24: rat] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X24 @ Y24 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups73079841787564623at_rat @ H @ S2 ) @ ( groups73079841787564623at_rat @ G2 @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_7584_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S2: set_int,H: int > rat,G2: int > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X1: rat,Y1: rat,X24: rat,Y24: rat] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups1072433553688619179nt_rat @ H @ S2 ) @ ( groups1072433553688619179nt_rat @ G2 @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_7585_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G2: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1681761925125756287l_real @ G2 @ ( insert_real @ X @ A2 ) )
            = ( groups1681761925125756287l_real @ G2 @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1681761925125756287l_real @ G2 @ ( insert_real @ X @ A2 ) )
            = ( times_times_real @ ( G2 @ X ) @ ( groups1681761925125756287l_real @ G2 @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_7586_prod_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G2: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( groups234877984723959775o_real @ G2 @ ( insert_o @ X @ A2 ) )
            = ( groups234877984723959775o_real @ G2 @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( groups234877984723959775o_real @ G2 @ ( insert_o @ X @ A2 ) )
            = ( times_times_real @ ( G2 @ X ) @ ( groups234877984723959775o_real @ G2 @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_7587_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G2: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups129246275422532515t_real @ G2 @ ( insert_nat @ X @ A2 ) )
            = ( groups129246275422532515t_real @ G2 @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups129246275422532515t_real @ G2 @ ( insert_nat @ X @ A2 ) )
            = ( times_times_real @ ( G2 @ X ) @ ( groups129246275422532515t_real @ G2 @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_7588_prod_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G2: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups2316167850115554303t_real @ G2 @ ( insert_int @ X @ A2 ) )
            = ( groups2316167850115554303t_real @ G2 @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups2316167850115554303t_real @ G2 @ ( insert_int @ X @ A2 ) )
            = ( times_times_real @ ( G2 @ X ) @ ( groups2316167850115554303t_real @ G2 @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_7589_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups766887009212190081x_real @ G2 @ ( insert_complex @ X @ A2 ) )
            = ( groups766887009212190081x_real @ G2 @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups766887009212190081x_real @ G2 @ ( insert_complex @ X @ A2 ) )
            = ( times_times_real @ ( G2 @ X ) @ ( groups766887009212190081x_real @ G2 @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_7590_prod_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ X @ A2 )
         => ( ( groups97031904164794029t_real @ G2 @ ( insert_Extended_enat @ X @ A2 ) )
            = ( groups97031904164794029t_real @ G2 @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ X @ A2 )
         => ( ( groups97031904164794029t_real @ G2 @ ( insert_Extended_enat @ X @ A2 ) )
            = ( times_times_real @ ( G2 @ X ) @ ( groups97031904164794029t_real @ G2 @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_7591_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G2: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups4061424788464935467al_rat @ G2 @ ( insert_real @ X @ A2 ) )
            = ( groups4061424788464935467al_rat @ G2 @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups4061424788464935467al_rat @ G2 @ ( insert_real @ X @ A2 ) )
            = ( times_times_rat @ ( G2 @ X ) @ ( groups4061424788464935467al_rat @ G2 @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_7592_prod_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G2: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( groups2869687844427037835_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
            = ( groups2869687844427037835_o_rat @ G2 @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( groups2869687844427037835_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
            = ( times_times_rat @ ( G2 @ X ) @ ( groups2869687844427037835_o_rat @ G2 @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_7593_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
            = ( groups73079841787564623at_rat @ G2 @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
            = ( times_times_rat @ ( G2 @ X ) @ ( groups73079841787564623at_rat @ G2 @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_7594_prod_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G2: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
            = ( groups1072433553688619179nt_rat @ G2 @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
            = ( times_times_rat @ ( G2 @ X ) @ ( groups1072433553688619179nt_rat @ G2 @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_7595_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_o,S2: set_o,I: $o > $o,J: $o > $o,T3: set_o,G2: $o > complex,H: $o > complex] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [A5: $o] :
              ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
               => ( member_o @ ( J @ A5 ) @ ( minus_minus_set_o @ T3 @ T5 ) ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: $o] :
                    ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                   => ( member_o @ ( I @ B5 ) @ ( minus_minus_set_o @ S2 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = one_one_complex ) )
                 => ( ! [B5: $o] :
                        ( ( member_o @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = one_one_complex ) )
                   => ( ! [A5: $o] :
                          ( ( member_o @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups4859619685533338977omplex @ G2 @ S2 )
                        = ( groups4859619685533338977omplex @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7596_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_int,S2: set_o,I: int > $o,J: $o > int,T3: set_int,G2: $o > complex,H: int > complex] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A5: $o] :
              ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_o @ ( I @ B5 ) @ ( minus_minus_set_o @ S2 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = one_one_complex ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = one_one_complex ) )
                   => ( ! [A5: $o] :
                          ( ( member_o @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups4859619685533338977omplex @ G2 @ S2 )
                        = ( groups7440179247065528705omplex @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7597_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_complex,S2: set_o,I: complex > $o,J: $o > complex,T3: set_complex,G2: $o > complex,H: complex > complex] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A5: $o] :
              ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
               => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: complex] :
                    ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_o @ ( I @ B5 ) @ ( minus_minus_set_o @ S2 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = one_one_complex ) )
                 => ( ! [B5: complex] :
                        ( ( member_complex @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = one_one_complex ) )
                   => ( ! [A5: $o] :
                          ( ( member_o @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups4859619685533338977omplex @ G2 @ S2 )
                        = ( groups3708469109370488835omplex @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7598_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_Extended_enat,S2: set_o,I: extended_enat > $o,J: $o > extended_enat,T3: set_Extended_enat,G2: $o > complex,H: extended_enat > complex] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite4001608067531595151d_enat @ T5 )
       => ( ! [A5: $o] :
              ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
               => ( member_Extended_enat @ ( J @ A5 ) @ ( minus_925952699566721837d_enat @ T3 @ T5 ) ) )
           => ( ! [B5: extended_enat] :
                  ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: extended_enat] :
                    ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                   => ( member_o @ ( I @ B5 ) @ ( minus_minus_set_o @ S2 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = one_one_complex ) )
                 => ( ! [B5: extended_enat] :
                        ( ( member_Extended_enat @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = one_one_complex ) )
                   => ( ! [A5: $o] :
                          ( ( member_o @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups4859619685533338977omplex @ G2 @ S2 )
                        = ( groups4622424608036095791omplex @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7599_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_o,S2: set_int,I: $o > int,J: int > $o,T3: set_o,G2: int > complex,H: $o > complex] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_o @ ( J @ A5 ) @ ( minus_minus_set_o @ T3 @ T5 ) ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: $o] :
                    ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = one_one_complex ) )
                 => ( ! [B5: $o] :
                        ( ( member_o @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = one_one_complex ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups7440179247065528705omplex @ G2 @ S2 )
                        = ( groups4859619685533338977omplex @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7600_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G2: int > complex,H: int > complex] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = one_one_complex ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = one_one_complex ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups7440179247065528705omplex @ G2 @ S2 )
                        = ( groups7440179247065528705omplex @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7601_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G2: int > complex,H: complex > complex] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: complex] :
                    ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = one_one_complex ) )
                 => ( ! [B5: complex] :
                        ( ( member_complex @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = one_one_complex ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups7440179247065528705omplex @ G2 @ S2 )
                        = ( groups3708469109370488835omplex @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7602_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_Extended_enat,S2: set_int,I: extended_enat > int,J: int > extended_enat,T3: set_Extended_enat,G2: int > complex,H: extended_enat > complex] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite4001608067531595151d_enat @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_Extended_enat @ ( J @ A5 ) @ ( minus_925952699566721837d_enat @ T3 @ T5 ) ) )
           => ( ! [B5: extended_enat] :
                  ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: extended_enat] :
                    ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = one_one_complex ) )
                 => ( ! [B5: extended_enat] :
                        ( ( member_Extended_enat @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = one_one_complex ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups7440179247065528705omplex @ G2 @ S2 )
                        = ( groups4622424608036095791omplex @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7603_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_o,S2: set_complex,I: $o > complex,J: complex > $o,T3: set_o,G2: complex > complex,H: $o > complex] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [A5: complex] :
              ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_o @ ( J @ A5 ) @ ( minus_minus_set_o @ T3 @ T5 ) ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: $o] :
                    ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B5 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A5: complex] :
                      ( ( member_complex @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = one_one_complex ) )
                 => ( ! [B5: $o] :
                        ( ( member_o @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = one_one_complex ) )
                   => ( ! [A5: complex] :
                          ( ( member_complex @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups3708469109370488835omplex @ G2 @ S2 )
                        = ( groups4859619685533338977omplex @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7604_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_int,S2: set_complex,I: int > complex,J: complex > int,T3: set_int,G2: complex > complex,H: int > complex] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A5: complex] :
              ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B5 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A5: complex] :
                      ( ( member_complex @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = one_one_complex ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = one_one_complex ) )
                   => ( ! [A5: complex] :
                          ( ( member_complex @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups3708469109370488835omplex @ G2 @ S2 )
                        = ( groups7440179247065528705omplex @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7605_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G2: int > complex] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups7440179247065528705omplex @ G2
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G2 @ X3 )
                  = one_one_complex ) ) ) )
        = ( groups7440179247065528705omplex @ G2 @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7606_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G2
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G2 @ X3 )
                  = one_one_complex ) ) ) )
        = ( groups3708469109370488835omplex @ G2 @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7607_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4622424608036095791omplex @ G2
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X3: extended_enat] :
                  ( ( G2 @ X3 )
                  = one_one_complex ) ) ) )
        = ( groups4622424608036095791omplex @ G2 @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7608_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G2: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G2
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G2 @ X3 )
                  = one_one_real ) ) ) )
        = ( groups2316167850115554303t_real @ G2 @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7609_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G2
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G2 @ X3 )
                  = one_one_real ) ) ) )
        = ( groups766887009212190081x_real @ G2 @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7610_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups97031904164794029t_real @ G2
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X3: extended_enat] :
                  ( ( G2 @ X3 )
                  = one_one_real ) ) ) )
        = ( groups97031904164794029t_real @ G2 @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7611_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G2: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G2
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G2 @ X3 )
                  = one_one_rat ) ) ) )
        = ( groups1072433553688619179nt_rat @ G2 @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7612_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups225925009352817453ex_rat @ G2
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G2 @ X3 )
                  = one_one_rat ) ) ) )
        = ( groups225925009352817453ex_rat @ G2 @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7613_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2245840878043517529at_rat @ G2
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X3: extended_enat] :
                  ( ( G2 @ X3 )
                  = one_one_rat ) ) ) )
        = ( groups2245840878043517529at_rat @ G2 @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7614_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G2: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G2
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G2 @ X3 )
                  = one_one_nat ) ) ) )
        = ( groups1707563613775114915nt_nat @ G2 @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7615_less__1__prod2,axiom,
    ! [I5: set_o,I: $o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups234877984723959775o_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7616_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7617_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7618_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7619_less__1__prod2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( member_Extended_enat @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups97031904164794029t_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7620_less__1__prod2,axiom,
    ! [I5: set_o,I: $o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7621_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7622_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7623_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7624_less__1__prod2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( member_Extended_enat @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups2245840878043517529at_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7625_less__1__prod,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7626_less__1__prod,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups97031904164794029t_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7627_less__1__prod,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7628_less__1__prod,axiom,
    ! [I5: set_o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups234877984723959775o_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7629_less__1__prod,axiom,
    ! [I5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7630_less__1__prod,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7631_less__1__prod,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7632_less__1__prod,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups2245840878043517529at_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7633_less__1__prod,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7634_less__1__prod,axiom,
    ! [I5: set_o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7635_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G2: complex > real] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups766887009212190081x_real @ G2 @ A2 )
          = ( times_times_real @ ( groups766887009212190081x_real @ G2 @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups766887009212190081x_real @ G2 @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7636_prod_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G2: extended_enat > real] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups97031904164794029t_real @ G2 @ A2 )
          = ( times_times_real @ ( groups97031904164794029t_real @ G2 @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( groups97031904164794029t_real @ G2 @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7637_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G2: nat > real] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups129246275422532515t_real @ G2 @ A2 )
          = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups129246275422532515t_real @ G2 @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7638_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G2: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups225925009352817453ex_rat @ G2 @ A2 )
          = ( times_times_rat @ ( groups225925009352817453ex_rat @ G2 @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups225925009352817453ex_rat @ G2 @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7639_prod_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G2: extended_enat > rat] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2245840878043517529at_rat @ G2 @ A2 )
          = ( times_times_rat @ ( groups2245840878043517529at_rat @ G2 @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( groups2245840878043517529at_rat @ G2 @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7640_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G2: nat > rat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups73079841787564623at_rat @ G2 @ A2 )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups73079841787564623at_rat @ G2 @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7641_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G2: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups861055069439313189ex_nat @ G2 @ A2 )
          = ( times_times_nat @ ( groups861055069439313189ex_nat @ G2 @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups861055069439313189ex_nat @ G2 @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7642_prod_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G2: extended_enat > nat] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2880970938130013265at_nat @ G2 @ A2 )
          = ( times_times_nat @ ( groups2880970938130013265at_nat @ G2 @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( groups2880970938130013265at_nat @ G2 @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7643_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G2: complex > int] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups858564598930262913ex_int @ G2 @ A2 )
          = ( times_times_int @ ( groups858564598930262913ex_int @ G2 @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups858564598930262913ex_int @ G2 @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7644_prod_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G2: extended_enat > int] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2878480467620962989at_int @ G2 @ A2 )
          = ( times_times_int @ ( groups2878480467620962989at_int @ G2 @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( groups2878480467620962989at_int @ G2 @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7645_prod_Osame__carrier,axiom,
    ! [C2: set_o,A2: set_o,B2: set_o,G2: $o > complex,H: $o > complex] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A2 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_complex ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_complex ) )
             => ( ( ( groups4859619685533338977omplex @ G2 @ A2 )
                  = ( groups4859619685533338977omplex @ H @ B2 ) )
                = ( ( groups4859619685533338977omplex @ G2 @ C2 )
                  = ( groups4859619685533338977omplex @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7646_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G2: complex > complex,H: complex > complex] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_complex ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G2 @ A2 )
                  = ( groups3708469109370488835omplex @ H @ B2 ) )
                = ( ( groups3708469109370488835omplex @ G2 @ C2 )
                  = ( groups3708469109370488835omplex @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7647_prod_Osame__carrier,axiom,
    ! [C2: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G2: extended_enat > complex,H: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ C2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C2 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_complex ) )
           => ( ! [B5: extended_enat] :
                  ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_complex ) )
             => ( ( ( groups4622424608036095791omplex @ G2 @ A2 )
                  = ( groups4622424608036095791omplex @ H @ B2 ) )
                = ( ( groups4622424608036095791omplex @ G2 @ C2 )
                  = ( groups4622424608036095791omplex @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7648_prod_Osame__carrier,axiom,
    ! [C2: set_o,A2: set_o,B2: set_o,G2: $o > real,H: $o > real] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A2 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_real ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_real ) )
             => ( ( ( groups234877984723959775o_real @ G2 @ A2 )
                  = ( groups234877984723959775o_real @ H @ B2 ) )
                = ( ( groups234877984723959775o_real @ G2 @ C2 )
                  = ( groups234877984723959775o_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7649_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G2: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_real ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G2 @ A2 )
                  = ( groups766887009212190081x_real @ H @ B2 ) )
                = ( ( groups766887009212190081x_real @ G2 @ C2 )
                  = ( groups766887009212190081x_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7650_prod_Osame__carrier,axiom,
    ! [C2: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G2: extended_enat > real,H: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C2 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_real ) )
           => ( ! [B5: extended_enat] :
                  ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_real ) )
             => ( ( ( groups97031904164794029t_real @ G2 @ A2 )
                  = ( groups97031904164794029t_real @ H @ B2 ) )
                = ( ( groups97031904164794029t_real @ G2 @ C2 )
                  = ( groups97031904164794029t_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7651_prod_Osame__carrier,axiom,
    ! [C2: set_o,A2: set_o,B2: set_o,G2: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A2 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_rat ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_rat ) )
             => ( ( ( groups2869687844427037835_o_rat @ G2 @ A2 )
                  = ( groups2869687844427037835_o_rat @ H @ B2 ) )
                = ( ( groups2869687844427037835_o_rat @ G2 @ C2 )
                  = ( groups2869687844427037835_o_rat @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7652_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G2: complex > rat,H: complex > rat] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_rat ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_rat ) )
             => ( ( ( groups225925009352817453ex_rat @ G2 @ A2 )
                  = ( groups225925009352817453ex_rat @ H @ B2 ) )
                = ( ( groups225925009352817453ex_rat @ G2 @ C2 )
                  = ( groups225925009352817453ex_rat @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7653_prod_Osame__carrier,axiom,
    ! [C2: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G2: extended_enat > rat,H: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ C2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C2 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_rat ) )
           => ( ! [B5: extended_enat] :
                  ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_rat ) )
             => ( ( ( groups2245840878043517529at_rat @ G2 @ A2 )
                  = ( groups2245840878043517529at_rat @ H @ B2 ) )
                = ( ( groups2245840878043517529at_rat @ G2 @ C2 )
                  = ( groups2245840878043517529at_rat @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7654_prod_Osame__carrier,axiom,
    ! [C2: set_o,A2: set_o,B2: set_o,G2: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A2 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_nat ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_nat ) )
             => ( ( ( groups3504817904513533571_o_nat @ G2 @ A2 )
                  = ( groups3504817904513533571_o_nat @ H @ B2 ) )
                = ( ( groups3504817904513533571_o_nat @ G2 @ C2 )
                  = ( groups3504817904513533571_o_nat @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7655_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A2: set_o,B2: set_o,G2: $o > complex,H: $o > complex] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A2 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_complex ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_complex ) )
             => ( ( ( groups4859619685533338977omplex @ G2 @ C2 )
                  = ( groups4859619685533338977omplex @ H @ C2 ) )
               => ( ( groups4859619685533338977omplex @ G2 @ A2 )
                  = ( groups4859619685533338977omplex @ H @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7656_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G2: complex > complex,H: complex > complex] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_complex ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G2 @ C2 )
                  = ( groups3708469109370488835omplex @ H @ C2 ) )
               => ( ( groups3708469109370488835omplex @ G2 @ A2 )
                  = ( groups3708469109370488835omplex @ H @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7657_prod_Osame__carrierI,axiom,
    ! [C2: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G2: extended_enat > complex,H: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ C2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C2 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_complex ) )
           => ( ! [B5: extended_enat] :
                  ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_complex ) )
             => ( ( ( groups4622424608036095791omplex @ G2 @ C2 )
                  = ( groups4622424608036095791omplex @ H @ C2 ) )
               => ( ( groups4622424608036095791omplex @ G2 @ A2 )
                  = ( groups4622424608036095791omplex @ H @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7658_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A2: set_o,B2: set_o,G2: $o > real,H: $o > real] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A2 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_real ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_real ) )
             => ( ( ( groups234877984723959775o_real @ G2 @ C2 )
                  = ( groups234877984723959775o_real @ H @ C2 ) )
               => ( ( groups234877984723959775o_real @ G2 @ A2 )
                  = ( groups234877984723959775o_real @ H @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7659_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G2: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_real ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G2 @ C2 )
                  = ( groups766887009212190081x_real @ H @ C2 ) )
               => ( ( groups766887009212190081x_real @ G2 @ A2 )
                  = ( groups766887009212190081x_real @ H @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7660_prod_Osame__carrierI,axiom,
    ! [C2: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G2: extended_enat > real,H: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C2 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_real ) )
           => ( ! [B5: extended_enat] :
                  ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_real ) )
             => ( ( ( groups97031904164794029t_real @ G2 @ C2 )
                  = ( groups97031904164794029t_real @ H @ C2 ) )
               => ( ( groups97031904164794029t_real @ G2 @ A2 )
                  = ( groups97031904164794029t_real @ H @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7661_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A2: set_o,B2: set_o,G2: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A2 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_rat ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_rat ) )
             => ( ( ( groups2869687844427037835_o_rat @ G2 @ C2 )
                  = ( groups2869687844427037835_o_rat @ H @ C2 ) )
               => ( ( groups2869687844427037835_o_rat @ G2 @ A2 )
                  = ( groups2869687844427037835_o_rat @ H @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7662_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B2: set_complex,G2: complex > rat,H: complex > rat] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_rat ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_rat ) )
             => ( ( ( groups225925009352817453ex_rat @ G2 @ C2 )
                  = ( groups225925009352817453ex_rat @ H @ C2 ) )
               => ( ( groups225925009352817453ex_rat @ G2 @ A2 )
                  = ( groups225925009352817453ex_rat @ H @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7663_prod_Osame__carrierI,axiom,
    ! [C2: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G2: extended_enat > rat,H: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ C2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C2 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_rat ) )
           => ( ! [B5: extended_enat] :
                  ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_rat ) )
             => ( ( ( groups2245840878043517529at_rat @ G2 @ C2 )
                  = ( groups2245840878043517529at_rat @ H @ C2 ) )
               => ( ( groups2245840878043517529at_rat @ G2 @ A2 )
                  = ( groups2245840878043517529at_rat @ H @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7664_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A2: set_o,B2: set_o,G2: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A2 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ C2 @ A2 ) )
               => ( ( G2 @ A5 )
                  = one_one_nat ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H @ B5 )
                    = one_one_nat ) )
             => ( ( ( groups3504817904513533571_o_nat @ G2 @ C2 )
                  = ( groups3504817904513533571_o_nat @ H @ C2 ) )
               => ( ( groups3504817904513533571_o_nat @ G2 @ A2 )
                  = ( groups3504817904513533571_o_nat @ H @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7665_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G2 @ S2 )
            = ( groups3708469109370488835omplex @ G2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7666_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_complex ) )
         => ( ( groups4622424608036095791omplex @ G2 @ S2 )
            = ( groups4622424608036095791omplex @ G2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7667_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G2 @ S2 )
            = ( groups766887009212190081x_real @ G2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7668_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_real ) )
         => ( ( groups97031904164794029t_real @ G2 @ S2 )
            = ( groups97031904164794029t_real @ G2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7669_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G2 @ S2 )
            = ( groups225925009352817453ex_rat @ G2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7670_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_rat ) )
         => ( ( groups2245840878043517529at_rat @ G2 @ S2 )
            = ( groups2245840878043517529at_rat @ G2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7671_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G2 @ S2 )
            = ( groups861055069439313189ex_nat @ G2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7672_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_nat ) )
         => ( ( groups2880970938130013265at_nat @ G2 @ S2 )
            = ( groups2880970938130013265at_nat @ G2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7673_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G2 @ S2 )
            = ( groups858564598930262913ex_int @ G2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7674_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_int ) )
         => ( ( groups2878480467620962989at_int @ G2 @ S2 )
            = ( groups2878480467620962989at_int @ G2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7675_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G2 @ T3 )
            = ( groups3708469109370488835omplex @ G2 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7676_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_complex ) )
         => ( ( groups4622424608036095791omplex @ G2 @ T3 )
            = ( groups4622424608036095791omplex @ G2 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7677_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G2 @ T3 )
            = ( groups766887009212190081x_real @ G2 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7678_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_real ) )
         => ( ( groups97031904164794029t_real @ G2 @ T3 )
            = ( groups97031904164794029t_real @ G2 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7679_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G2 @ T3 )
            = ( groups225925009352817453ex_rat @ G2 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7680_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_rat ) )
         => ( ( groups2245840878043517529at_rat @ G2 @ T3 )
            = ( groups2245840878043517529at_rat @ G2 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7681_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G2 @ T3 )
            = ( groups861055069439313189ex_nat @ G2 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7682_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_nat ) )
         => ( ( groups2880970938130013265at_nat @ G2 @ T3 )
            = ( groups2880970938130013265at_nat @ G2 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7683_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G2 @ T3 )
            = ( groups858564598930262913ex_int @ G2 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7684_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_int ) )
         => ( ( groups2878480467620962989at_int @ G2 @ T3 )
            = ( groups2878480467620962989at_int @ G2 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7685_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S2: set_o,H: $o > complex,G2: $o > complex] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S2 @ T3 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T3 @ S2 ) )
             => ( ( H @ X4 )
                = one_one_complex ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups4859619685533338977omplex @ G2 @ S2 )
              = ( groups4859619685533338977omplex @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7686_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: complex > complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H @ X4 )
                = one_one_complex ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups3708469109370488835omplex @ G2 @ S2 )
              = ( groups3708469109370488835omplex @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7687_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H: extended_enat > complex,G2: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( H @ X4 )
                = one_one_complex ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups4622424608036095791omplex @ G2 @ S2 )
              = ( groups4622424608036095791omplex @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7688_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S2: set_o,H: $o > real,G2: $o > real] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S2 @ T3 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T3 @ S2 ) )
             => ( ( H @ X4 )
                = one_one_real ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups234877984723959775o_real @ G2 @ S2 )
              = ( groups234877984723959775o_real @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7689_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: complex > real,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H @ X4 )
                = one_one_real ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups766887009212190081x_real @ G2 @ S2 )
              = ( groups766887009212190081x_real @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7690_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H: extended_enat > real,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( H @ X4 )
                = one_one_real ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups97031904164794029t_real @ G2 @ S2 )
              = ( groups97031904164794029t_real @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7691_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S2: set_o,H: $o > rat,G2: $o > rat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S2 @ T3 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T3 @ S2 ) )
             => ( ( H @ X4 )
                = one_one_rat ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups2869687844427037835_o_rat @ G2 @ S2 )
              = ( groups2869687844427037835_o_rat @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7692_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H: complex > rat,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H @ X4 )
                = one_one_rat ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups225925009352817453ex_rat @ G2 @ S2 )
              = ( groups225925009352817453ex_rat @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7693_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H: extended_enat > rat,G2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( H @ X4 )
                = one_one_rat ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups2245840878043517529at_rat @ G2 @ S2 )
              = ( groups2245840878043517529at_rat @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7694_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S2: set_o,H: $o > nat,G2: $o > nat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S2 @ T3 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T3 @ S2 ) )
             => ( ( H @ X4 )
                = one_one_nat ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups3504817904513533571_o_nat @ G2 @ S2 )
              = ( groups3504817904513533571_o_nat @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7695_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S2: set_o,G2: $o > complex,H: $o > complex] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S2 @ T3 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_complex ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups4859619685533338977omplex @ G2 @ T3 )
              = ( groups4859619685533338977omplex @ H @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7696_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > complex,H: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_complex ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups3708469109370488835omplex @ G2 @ T3 )
              = ( groups3708469109370488835omplex @ H @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7697_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > complex,H: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_complex ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups4622424608036095791omplex @ G2 @ T3 )
              = ( groups4622424608036095791omplex @ H @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7698_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S2: set_o,G2: $o > real,H: $o > real] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S2 @ T3 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_real ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups234877984723959775o_real @ G2 @ T3 )
              = ( groups234877984723959775o_real @ H @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7699_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_real ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups766887009212190081x_real @ G2 @ T3 )
              = ( groups766887009212190081x_real @ H @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7700_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > real,H: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_real ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups97031904164794029t_real @ G2 @ T3 )
              = ( groups97031904164794029t_real @ H @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7701_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S2: set_o,G2: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S2 @ T3 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_rat ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups2869687844427037835_o_rat @ G2 @ T3 )
              = ( groups2869687844427037835_o_rat @ H @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7702_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G2: complex > rat,H: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_rat ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups225925009352817453ex_rat @ G2 @ T3 )
              = ( groups225925009352817453ex_rat @ H @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7703_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G2: extended_enat > rat,H: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_rat ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups2245840878043517529at_rat @ G2 @ T3 )
              = ( groups2245840878043517529at_rat @ H @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7704_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S2: set_o,G2: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S2 @ T3 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T3 @ S2 ) )
             => ( ( G2 @ X4 )
                = one_one_nat ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G2 @ X4 )
                  = ( H @ X4 ) ) )
           => ( ( groups3504817904513533571_o_nat @ G2 @ T3 )
              = ( groups3504817904513533571_o_nat @ H @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7705_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > real,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G2 @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7706_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => ( ord_less_real @ ( groups97031904164794029t_real @ F @ A2 ) @ ( groups97031904164794029t_real @ G2 @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7707_prod__mono__strict,axiom,
    ! [A2: set_real,F: real > real,G2: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_real )
         => ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G2 @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7708_prod__mono__strict,axiom,
    ! [A2: set_o,F: $o > real,G2: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_o )
         => ( ord_less_real @ ( groups234877984723959775o_real @ F @ A2 ) @ ( groups234877984723959775o_real @ G2 @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7709_prod__mono__strict,axiom,
    ! [A2: set_nat,F: nat > real,G2: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_nat )
         => ( ord_less_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G2 @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7710_prod__mono__strict,axiom,
    ! [A2: set_int,F: int > real,G2: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_int )
         => ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G2 @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7711_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > rat,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G2 @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7712_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > rat,G2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => ( ord_less_rat @ ( groups2245840878043517529at_rat @ F @ A2 ) @ ( groups2245840878043517529at_rat @ G2 @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7713_prod__mono__strict,axiom,
    ! [A2: set_real,F: real > rat,G2: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_real )
         => ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G2 @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7714_prod__mono__strict,axiom,
    ! [A2: set_o,F: $o > rat,G2: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_o )
         => ( ord_less_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) @ ( groups2869687844427037835_o_rat @ G2 @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7715_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G2: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G2 @ ( insert_complex @ X @ A2 ) )
        = ( times_times_real @ ( G2 @ X ) @ ( groups766887009212190081x_real @ G2 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7716_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > real,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups97031904164794029t_real @ G2 @ ( insert_Extended_enat @ X @ A2 ) )
        = ( times_times_real @ ( G2 @ X ) @ ( groups97031904164794029t_real @ G2 @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7717_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G2: real > real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1681761925125756287l_real @ G2 @ ( insert_real @ X @ A2 ) )
        = ( times_times_real @ ( G2 @ X ) @ ( groups1681761925125756287l_real @ G2 @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7718_prod_Oinsert__remove,axiom,
    ! [A2: set_o,G2: $o > real,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups234877984723959775o_real @ G2 @ ( insert_o @ X @ A2 ) )
        = ( times_times_real @ ( G2 @ X ) @ ( groups234877984723959775o_real @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7719_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G2: int > real,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G2 @ ( insert_int @ X @ A2 ) )
        = ( times_times_real @ ( G2 @ X ) @ ( groups2316167850115554303t_real @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7720_prod_Oinsert__remove,axiom,
    ! [A2: set_nat,G2: nat > real,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G2 @ ( insert_nat @ X @ A2 ) )
        = ( times_times_real @ ( G2 @ X ) @ ( groups129246275422532515t_real @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7721_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G2: complex > rat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups225925009352817453ex_rat @ G2 @ ( insert_complex @ X @ A2 ) )
        = ( times_times_rat @ ( G2 @ X ) @ ( groups225925009352817453ex_rat @ G2 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7722_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > rat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2245840878043517529at_rat @ G2 @ ( insert_Extended_enat @ X @ A2 ) )
        = ( times_times_rat @ ( G2 @ X ) @ ( groups2245840878043517529at_rat @ G2 @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7723_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G2: real > rat,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4061424788464935467al_rat @ G2 @ ( insert_real @ X @ A2 ) )
        = ( times_times_rat @ ( G2 @ X ) @ ( groups4061424788464935467al_rat @ G2 @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7724_prod_Oinsert__remove,axiom,
    ! [A2: set_o,G2: $o > rat,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups2869687844427037835_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
        = ( times_times_rat @ ( G2 @ X ) @ ( groups2869687844427037835_o_rat @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7725_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups766887009212190081x_real @ G2 @ A2 )
          = ( times_times_real @ ( G2 @ X ) @ ( groups766887009212190081x_real @ G2 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7726_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups97031904164794029t_real @ G2 @ A2 )
          = ( times_times_real @ ( G2 @ X ) @ ( groups97031904164794029t_real @ G2 @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7727_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G2: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups1681761925125756287l_real @ G2 @ A2 )
          = ( times_times_real @ ( G2 @ X ) @ ( groups1681761925125756287l_real @ G2 @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7728_prod_Oremove,axiom,
    ! [A2: set_o,X: $o,G2: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ( groups234877984723959775o_real @ G2 @ A2 )
          = ( times_times_real @ ( G2 @ X ) @ ( groups234877984723959775o_real @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7729_prod_Oremove,axiom,
    ! [A2: set_int,X: int,G2: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups2316167850115554303t_real @ G2 @ A2 )
          = ( times_times_real @ ( G2 @ X ) @ ( groups2316167850115554303t_real @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7730_prod_Oremove,axiom,
    ! [A2: set_nat,X: nat,G2: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( groups129246275422532515t_real @ G2 @ A2 )
          = ( times_times_real @ ( G2 @ X ) @ ( groups129246275422532515t_real @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7731_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups225925009352817453ex_rat @ G2 @ A2 )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups225925009352817453ex_rat @ G2 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7732_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2245840878043517529at_rat @ G2 @ A2 )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups2245840878043517529at_rat @ G2 @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7733_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G2: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups4061424788464935467al_rat @ G2 @ A2 )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups4061424788464935467al_rat @ G2 @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7734_prod_Oremove,axiom,
    ! [A2: set_o,X: $o,G2: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ( groups2869687844427037835_o_rat @ G2 @ A2 )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups2869687844427037835_o_rat @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7735_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( groups766887009212190081x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups766887009212190081x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7736_prod_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > real,C: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( groups97031904164794029t_real @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups97031904164794029t_real @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7737_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: real > real,C: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( groups1681761925125756287l_real @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups1681761925125756287l_real @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7738_prod_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: $o > real,C: $o > real] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups234877984723959775o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( groups234877984723959775o_real @ C @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups234877984723959775o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups234877984723959775o_real @ C @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7739_prod_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: int > real,C: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( groups2316167850115554303t_real @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups2316167850115554303t_real @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7740_prod_Odelta__remove,axiom,
    ! [S2: set_nat,A: nat,B: nat > real,C: nat > real] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( groups129246275422532515t_real @ C @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups129246275422532515t_real @ C @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7741_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups225925009352817453ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( times_times_rat @ ( B @ A ) @ ( groups225925009352817453ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups225925009352817453ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups225925009352817453ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7742_prod_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > rat,C: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2245840878043517529at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( times_times_rat @ ( B @ A ) @ ( groups2245840878043517529at_rat @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2245840878043517529at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups2245840878043517529at_rat @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7743_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: real > rat,C: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( times_times_rat @ ( B @ A ) @ ( groups4061424788464935467al_rat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups4061424788464935467al_rat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7744_prod_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: $o > rat,C: $o > rat] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( times_times_rat @ ( B @ A ) @ ( groups2869687844427037835_o_rat @ C @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups2869687844427037835_o_rat @ C @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7745_prod__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: $o > real] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ! [B5: $o] :
              ( ( member_o @ B5 @ ( minus_minus_set_o @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B5 ) ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A5 ) ) )
           => ( ord_less_eq_real @ ( groups234877984723959775o_real @ F @ A2 ) @ ( groups234877984723959775o_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7746_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B5 ) ) )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A5 ) ) )
           => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7747_prod__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ! [B5: extended_enat] :
              ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B5 ) ) )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A5 ) ) )
           => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A2 ) @ ( groups97031904164794029t_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7748_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ! [B5: nat] :
              ( ( member_nat @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B5 ) ) )
         => ( ! [A5: nat] :
                ( ( member_nat @ A5 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A5 ) ) )
           => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7749_prod__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: $o > rat] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ! [B5: $o] :
              ( ( member_o @ B5 @ ( minus_minus_set_o @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B5 ) ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A5 ) ) )
           => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) @ ( groups2869687844427037835_o_rat @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7750_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B5 ) ) )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A5 ) ) )
           => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7751_prod__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ! [B5: extended_enat] :
              ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B5 ) ) )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A5 ) ) )
           => ( ord_less_eq_rat @ ( groups2245840878043517529at_rat @ F @ A2 ) @ ( groups2245840878043517529at_rat @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7752_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ! [B5: nat] :
              ( ( member_nat @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B5 ) ) )
         => ( ! [A5: nat] :
                ( ( member_nat @ A5 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A5 ) ) )
           => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7753_prod__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: $o > int] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ! [B5: $o] :
              ( ( member_o @ B5 @ ( minus_minus_set_o @ B2 @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B5 ) ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A5 ) ) )
           => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A2 ) @ ( groups3502327434004483295_o_int @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7754_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B5 ) ) )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A5 ) ) )
           => ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7755_prod__le__power,axiom,
    ! [A2: set_o,F: $o > real,N: real,K: nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ K )
       => ( ( ord_less_eq_real @ one_one_real @ N )
         => ( ord_less_eq_real @ ( groups234877984723959775o_real @ F @ A2 ) @ ( power_power_real @ N @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_7756_prod__le__power,axiom,
    ! [A2: set_complex,F: complex > real,N: real,K: nat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ K )
       => ( ( ord_less_eq_real @ one_one_real @ N )
         => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( power_power_real @ N @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_7757_prod__le__power,axiom,
    ! [A2: set_nat,F: nat > real,N: real,K: nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ K )
       => ( ( ord_less_eq_real @ one_one_real @ N )
         => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( power_power_real @ N @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_7758_prod__le__power,axiom,
    ! [A2: set_int,F: int > real,N: real,K: nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ K )
       => ( ( ord_less_eq_real @ one_one_real @ N )
         => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( power_power_real @ N @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_7759_prod__le__power,axiom,
    ! [A2: set_o,F: $o > rat,N: rat,K: nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ K )
       => ( ( ord_less_eq_rat @ one_one_rat @ N )
         => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) @ ( power_power_rat @ N @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_7760_prod__le__power,axiom,
    ! [A2: set_complex,F: complex > rat,N: rat,K: nat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ K )
       => ( ( ord_less_eq_rat @ one_one_rat @ N )
         => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( power_power_rat @ N @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_7761_prod__le__power,axiom,
    ! [A2: set_nat,F: nat > rat,N: rat,K: nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ K )
       => ( ( ord_less_eq_rat @ one_one_rat @ N )
         => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( power_power_rat @ N @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_7762_prod__le__power,axiom,
    ! [A2: set_int,F: int > rat,N: rat,K: nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ K )
       => ( ( ord_less_eq_rat @ one_one_rat @ N )
         => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( power_power_rat @ N @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_7763_prod__le__power,axiom,
    ! [A2: set_o,F: $o > nat,N: nat,K: nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ K )
       => ( ( ord_less_eq_nat @ one_one_nat @ N )
         => ( ord_less_eq_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) @ ( power_power_nat @ N @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_7764_prod__le__power,axiom,
    ! [A2: set_complex,F: complex > nat,N: nat,K: nat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ K )
       => ( ( ord_less_eq_nat @ one_one_nat @ N )
         => ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( power_power_nat @ N @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_7765_prod__diff1,axiom,
    ! [A2: set_complex,F: complex > rat,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( F @ A )
         != zero_zero_rat )
       => ( ( ( member_complex @ A @ A2 )
           => ( ( groups225925009352817453ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide_divide_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A2 )
           => ( ( groups225925009352817453ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups225925009352817453ex_rat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7766_prod__diff1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > rat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( F @ A )
         != zero_zero_rat )
       => ( ( ( member_Extended_enat @ A @ A2 )
           => ( ( groups2245840878043517529at_rat @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( divide_divide_rat @ ( groups2245840878043517529at_rat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_Extended_enat @ A @ A2 )
           => ( ( groups2245840878043517529at_rat @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( groups2245840878043517529at_rat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7767_prod__diff1,axiom,
    ! [A2: set_real,F: real > rat,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( F @ A )
         != zero_zero_rat )
       => ( ( ( member_real @ A @ A2 )
           => ( ( groups4061424788464935467al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( divide_divide_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_real @ A @ A2 )
           => ( ( groups4061424788464935467al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7768_prod__diff1,axiom,
    ! [A2: set_o,F: $o > rat,A: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( F @ A )
         != zero_zero_rat )
       => ( ( ( member_o @ A @ A2 )
           => ( ( groups2869687844427037835_o_rat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
              = ( divide_divide_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_o @ A @ A2 )
           => ( ( groups2869687844427037835_o_rat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
              = ( groups2869687844427037835_o_rat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7769_prod__diff1,axiom,
    ! [A2: set_int,F: int > rat,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( F @ A )
         != zero_zero_rat )
       => ( ( ( member_int @ A @ A2 )
           => ( ( groups1072433553688619179nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( divide_divide_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_int @ A @ A2 )
           => ( ( groups1072433553688619179nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7770_prod__diff1,axiom,
    ! [A2: set_nat,F: nat > rat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( F @ A )
         != zero_zero_rat )
       => ( ( ( member_nat @ A @ A2 )
           => ( ( groups73079841787564623at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( divide_divide_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_nat @ A @ A2 )
           => ( ( groups73079841787564623at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( groups73079841787564623at_rat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7771_prod__diff1,axiom,
    ! [A2: set_complex,F: complex > int,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( F @ A )
         != zero_zero_int )
       => ( ( ( member_complex @ A @ A2 )
           => ( ( groups858564598930262913ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide_divide_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A2 )
           => ( ( groups858564598930262913ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7772_prod__diff1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > int,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( F @ A )
         != zero_zero_int )
       => ( ( ( member_Extended_enat @ A @ A2 )
           => ( ( groups2878480467620962989at_int @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( divide_divide_int @ ( groups2878480467620962989at_int @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_Extended_enat @ A @ A2 )
           => ( ( groups2878480467620962989at_int @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( groups2878480467620962989at_int @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7773_prod__diff1,axiom,
    ! [A2: set_real,F: real > int,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( F @ A )
         != zero_zero_int )
       => ( ( ( member_real @ A @ A2 )
           => ( ( groups4694064378042380927al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( divide_divide_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_real @ A @ A2 )
           => ( ( groups4694064378042380927al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7774_prod__diff1,axiom,
    ! [A2: set_o,F: $o > int,A: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( F @ A )
         != zero_zero_int )
       => ( ( ( member_o @ A @ A2 )
           => ( ( groups3502327434004483295_o_int @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
              = ( divide_divide_int @ ( groups3502327434004483295_o_int @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_o @ A @ A2 )
           => ( ( groups3502327434004483295_o_int @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
              = ( groups3502327434004483295_o_int @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7775_prod__gen__delta,axiom,
    ! [S2: set_o,A: $o,B: $o > complex,C: complex] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups4859619685533338977omplex
              @ ^ [K3: $o] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( times_times_complex @ ( B @ A ) @ ( power_power_complex @ C @ ( minus_minus_nat @ ( finite_card_o @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups4859619685533338977omplex
              @ ^ [K3: $o] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( power_power_complex @ C @ ( finite_card_o @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_7776_prod__gen__delta,axiom,
    ! [S2: set_nat,A: nat,B: nat > complex,C: complex] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( times_times_complex @ ( B @ A ) @ ( power_power_complex @ C @ ( minus_minus_nat @ ( finite_card_nat @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( power_power_complex @ C @ ( finite_card_nat @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_7777_prod__gen__delta,axiom,
    ! [S2: set_int,A: int,B: int > complex,C: complex] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( times_times_complex @ ( B @ A ) @ ( power_power_complex @ C @ ( minus_minus_nat @ ( finite_card_int @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( power_power_complex @ C @ ( finite_card_int @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_7778_prod__gen__delta,axiom,
    ! [S2: set_complex,A: complex,B: complex > complex,C: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( times_times_complex @ ( B @ A ) @ ( power_power_complex @ C @ ( minus_minus_nat @ ( finite_card_complex @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( power_power_complex @ C @ ( finite_card_complex @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_7779_prod__gen__delta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > complex,C: complex] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups4622424608036095791omplex
              @ ^ [K3: extended_enat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( times_times_complex @ ( B @ A ) @ ( power_power_complex @ C @ ( minus_minus_nat @ ( finite121521170596916366d_enat @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups4622424608036095791omplex
              @ ^ [K3: extended_enat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( power_power_complex @ C @ ( finite121521170596916366d_enat @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_7780_prod__gen__delta,axiom,
    ! [S2: set_o,A: $o,B: $o > real,C: real] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups234877984723959775o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( power_power_real @ C @ ( minus_minus_nat @ ( finite_card_o @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups234877984723959775o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( power_power_real @ C @ ( finite_card_o @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_7781_prod__gen__delta,axiom,
    ! [S2: set_nat,A: nat,B: nat > real,C: real] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( power_power_real @ C @ ( minus_minus_nat @ ( finite_card_nat @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( power_power_real @ C @ ( finite_card_nat @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_7782_prod__gen__delta,axiom,
    ! [S2: set_int,A: int,B: int > real,C: real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( power_power_real @ C @ ( minus_minus_nat @ ( finite_card_int @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( power_power_real @ C @ ( finite_card_int @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_7783_prod__gen__delta,axiom,
    ! [S2: set_complex,A: complex,B: complex > real,C: real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( power_power_real @ C @ ( minus_minus_nat @ ( finite_card_complex @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( power_power_real @ C @ ( finite_card_complex @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_7784_prod__gen__delta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > real,C: real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( times_times_real @ ( B @ A ) @ ( power_power_real @ C @ ( minus_minus_nat @ ( finite121521170596916366d_enat @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C )
              @ S2 )
            = ( power_power_real @ C @ ( finite121521170596916366d_enat @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_7785_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_7786_round__unique,axiom,
    ! [X: real,Y: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_7787_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_7788_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7789_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_7790_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7791_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_7792_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7793_summable__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_7794_round__altdef,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X3: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X3 ) ) @ ( archim7802044766580827645g_real @ X3 ) @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ).

% round_altdef
thf(fact_7795_round__altdef,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X3: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X3 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ).

% round_altdef
thf(fact_7796_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_7797_round__unique_H,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X )
        = N ) ) ).

% round_unique'
thf(fact_7798_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_7799_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_7800_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_7801_summable__single,axiom,
    ! [I: nat,F: nat > real] :
      ( summable_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_7802_summable__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( summable_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_7803_summable__single,axiom,
    ! [I: nat,F: nat > int] :
      ( summable_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_7804_summable__zero,axiom,
    ( summable_real
    @ ^ [N4: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_7805_summable__zero,axiom,
    ( summable_nat
    @ ^ [N4: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_7806_summable__zero,axiom,
    ( summable_int
    @ ^ [N4: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_7807_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_7808_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_7809_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_7810_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_7811_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7812_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_7813_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_7814_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) )
      = ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_7815_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_7816_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7817_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_7818_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7819_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_7820_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7821_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite
thf(fact_7822_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite
thf(fact_7823_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite
thf(fact_7824_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite_set
thf(fact_7825_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite_set
thf(fact_7826_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite_set
thf(fact_7827_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7828_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7829_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7830_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7831_dbl__simps_I3_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7832_summable__comparison__test,axiom,
    ! [F: nat > real,G2: nat > real] :
      ( ? [N8: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N8 @ N2 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G2 @ N2 ) ) )
     => ( ( summable_real @ G2 )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_7833_summable__comparison__test,axiom,
    ! [F: nat > complex,G2: nat > real] :
      ( ? [N8: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N8 @ N2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G2 @ N2 ) ) )
     => ( ( summable_real @ G2 )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_7834_summable__comparison__test_H,axiom,
    ! [G2: nat > real,N5: nat,F: nat > real] :
      ( ( summable_real @ G2 )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G2 @ N2 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7835_summable__comparison__test_H,axiom,
    ! [G2: nat > real,N5: nat,F: nat > complex] :
      ( ( summable_real @ G2 )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G2 @ N2 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7836_summable__const__iff,axiom,
    ! [C: real] :
      ( ( summable_real
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_7837_suminf__le,axiom,
    ! [F: nat > real,G2: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G2 @ N2 ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G2 )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G2 ) ) ) ) ) ).

% suminf_le
thf(fact_7838_suminf__le,axiom,
    ! [F: nat > nat,G2: nat > nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G2 @ N2 ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G2 )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G2 ) ) ) ) ) ).

% suminf_le
thf(fact_7839_suminf__le,axiom,
    ! [F: nat > int,G2: nat > int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G2 @ N2 ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G2 )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G2 ) ) ) ) ) ).

% suminf_le
thf(fact_7840_summable__finite,axiom,
    ! [N5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_real ) )
       => ( summable_real @ F ) ) ) ).

% summable_finite
thf(fact_7841_summable__finite,axiom,
    ! [N5: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_nat ) )
       => ( summable_nat @ F ) ) ) ).

% summable_finite
thf(fact_7842_summable__finite,axiom,
    ! [N5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_int ) )
       => ( summable_int @ F ) ) ) ).

% summable_finite
thf(fact_7843_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_7844_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7845_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7846_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7847_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_7848_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X3: real] : ( plus_plus_real @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_7849_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_7850_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_7851_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N4: nat] :
                ( ( F @ N4 )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7852_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N4: nat] :
                ( ( F @ N4 )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7853_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N4: nat] :
                ( ( F @ N4 )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7854_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7855_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7856_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7857_suminf__pos,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_pos
thf(fact_7858_suminf__pos,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_pos
thf(fact_7859_suminf__pos,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_pos
thf(fact_7860_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) ) ).

% summable_zero_power'
thf(fact_7861_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) ) ).

% summable_zero_power'
thf(fact_7862_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N4: nat] : ( times_times_int @ ( F @ N4 ) @ ( power_power_int @ zero_zero_int @ N4 ) ) ) ).

% summable_zero_power'
thf(fact_7863_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) ) ).

% summable_0_powser
thf(fact_7864_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) ) ).

% summable_0_powser
thf(fact_7865_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).

% round_mono
thf(fact_7866_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G2: nat > real] :
      ( ? [N8: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N8 @ N2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G2 @ N2 ) ) )
     => ( ( summable_real @ G2 )
       => ( summable_real
          @ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_7867_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G2: nat > real] :
      ( ? [N8: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N8 @ N2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G2 @ N2 ) ) )
     => ( ( summable_real @ G2 )
       => ( summable_real
          @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_7868_floor__le__round,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).

% floor_le_round
thf(fact_7869_floor__le__round,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).

% floor_le_round
thf(fact_7870_ceiling__ge__round,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).

% ceiling_ge_round
thf(fact_7871_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).

% summable_rabs
thf(fact_7872_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7873_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7874_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7875_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7876_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7877_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7878_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7879_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_7880_summable__norm,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( F @ N4 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( F @ N4 ) ) ) ) ) ).

% summable_norm
thf(fact_7881_summable__norm,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
        @ ( suminf_real
          @ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) ) ).

% summable_norm
thf(fact_7882_pi__half__le__two,axiom,
    ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_le_two
thf(fact_7883_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
     => ( ( suminf_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7884_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7885_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7886_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7887_suminf__exist__split,axiom,
    ! [R2: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_real @ F )
       => ? [N9: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ N9 @ N6 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N6 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_7888_suminf__exist__split,axiom,
    ! [R2: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_complex @ F )
       => ? [N9: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ N9 @ N6 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N6 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_7889_round__diff__minimal,axiom,
    ! [Z: real,M2: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M2 ) ) ) ) ).

% round_diff_minimal
thf(fact_7890_round__diff__minimal,axiom,
    ! [Z: rat,M2: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M2 ) ) ) ) ).

% round_diff_minimal
thf(fact_7891_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_7892_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: nat > complex,M5: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ( ord_less_real @ R2 @ R0 )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R0 @ N2 ) ) @ M5 )
         => ( summable_real
            @ ^ [N4: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N4 ) ) @ ( power_power_real @ R2 @ N4 ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_7893_pi__half__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_ge_zero
thf(fact_7894_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_7895_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_7896_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X3: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_7897_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X3: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_7898_of__int__round__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7899_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_7900_of__int__round__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_ge
thf(fact_7901_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_7902_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_7903_of__int__round__gt,axiom,
    ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_gt
thf(fact_7904_of__int__round__abs__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7905_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_7906_card__lists__distinct__length__eq,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( ord_less_eq_nat @ K @ ( finite_card_list_nat @ A2 ) )
       => ( ( finite7325466520557071688st_nat
            @ ( collec5989764272469232197st_nat
              @ ^ [Xs2: list_list_nat] :
                  ( ( ( size_s3023201423986296836st_nat @ Xs2 )
                    = K )
                  & ( distinct_list_nat @ Xs2 )
                  & ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A2 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ K ) @ one_one_nat ) @ ( finite_card_list_nat @ A2 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_7907_card__lists__distinct__length__eq,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( ord_less_eq_nat @ K @ ( finite_card_set_nat @ A2 ) )
       => ( ( finite5631907774883551598et_nat
            @ ( collect_list_set_nat
              @ ^ [Xs2: list_set_nat] :
                  ( ( ( size_s3254054031482475050et_nat @ Xs2 )
                    = K )
                  & ( distinct_set_nat @ Xs2 )
                  & ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A2 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ K ) @ one_one_nat ) @ ( finite_card_set_nat @ A2 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_7908_card__lists__distinct__length__eq,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_less_eq_nat @ K @ ( finite_card_complex @ A2 ) )
       => ( ( finite5120063068150530198omplex
            @ ( collect_list_complex
              @ ^ [Xs2: list_complex] :
                  ( ( ( size_s3451745648224563538omplex @ Xs2 )
                    = K )
                  & ( distinct_complex @ Xs2 )
                  & ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_complex @ A2 ) @ K ) @ one_one_nat ) @ ( finite_card_complex @ A2 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_7909_card__lists__distinct__length__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ord_less_eq_nat @ K @ ( finite711546835091564841at_nat @ A2 ) )
       => ( ( finite249151656366948015at_nat
            @ ( collec3343600615725829874at_nat
              @ ^ [Xs2: list_P6011104703257516679at_nat] :
                  ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                    = K )
                  & ( distin6923225563576452346at_nat @ Xs2 )
                  & ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite711546835091564841at_nat @ A2 ) @ K ) @ one_one_nat ) @ ( finite711546835091564841at_nat @ A2 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_7910_card__lists__distinct__length__eq,axiom,
    ! [A2: set_Extended_enat,K: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ord_less_eq_nat @ K @ ( finite121521170596916366d_enat @ A2 ) )
       => ( ( finite7441382602597825044d_enat
            @ ( collec8433460942617342167d_enat
              @ ^ [Xs2: list_Extended_enat] :
                  ( ( ( size_s3941691890525107288d_enat @ Xs2 )
                    = K )
                  & ( distin4523846830085650399d_enat @ Xs2 )
                  & ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A2 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite121521170596916366d_enat @ A2 ) @ K ) @ one_one_nat ) @ ( finite121521170596916366d_enat @ A2 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_7911_card__lists__distinct__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,K: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ord_less_eq_nat @ K @ ( finite7802652506058667612T_VEBT @ A2 ) )
       => ( ( finite5915292604075114978T_VEBT
            @ ( collec5608196760682091941T_VEBT
              @ ^ [Xs2: list_VEBT_VEBT] :
                  ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                    = K )
                  & ( distinct_VEBT_VEBT @ Xs2 )
                  & ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite7802652506058667612T_VEBT @ A2 ) @ K ) @ one_one_nat ) @ ( finite7802652506058667612T_VEBT @ A2 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_7912_card__lists__distinct__length__eq,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_nat @ K @ ( finite_card_nat @ A2 ) )
       => ( ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [Xs2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = K )
                  & ( distinct_nat @ Xs2 )
                  & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ K ) @ one_one_nat ) @ ( finite_card_nat @ A2 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_7913_card__lists__distinct__length__eq,axiom,
    ! [A2: set_int,K: nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_nat @ K @ ( finite_card_int @ A2 ) )
       => ( ( finite_card_list_int
            @ ( collect_list_int
              @ ^ [Xs2: list_int] :
                  ( ( ( size_size_list_int @ Xs2 )
                    = K )
                  & ( distinct_int @ Xs2 )
                  & ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_int @ A2 ) @ K ) @ one_one_nat ) @ ( finite_card_int @ A2 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_7914_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_7915_binomial__code,axiom,
    ( binomial
    = ( ^ [N4: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K3 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_7916_accp__subset,axiom,
    ! [R1: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o,R22: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o] :
      ( ( ord_le1077754993875142464_nat_o @ R1 @ R22 )
     => ( ord_le7812727212727832188_nat_o @ ( accp_P2887432264394892906BT_nat @ R22 ) @ ( accp_P2887432264394892906BT_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7917_accp__subset,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
     => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7918_accp__subset,axiom,
    ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
      ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
     => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).

% accp_subset
thf(fact_7919_accp__subset,axiom,
    ! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
      ( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
     => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7920_accp__subset,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
     => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7921_sum__gp,axiom,
    ! [N: nat,M2: nat,X: complex] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M2 ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7922_sum__gp,axiom,
    ! [N: nat,M2: nat,X: rat] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7923_sum__gp,axiom,
    ! [N: nat,M2: nat,X: real] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M2 ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7924_binomial__n__n,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ N )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_7925_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A2 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_7926_sum_Oneutral__const,axiom,
    ! [A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A2 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_7927_sum_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A2 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_7928_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A2 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_7929_sum_Oempty,axiom,
    ! [G2: real > real] :
      ( ( groups8097168146408367636l_real @ G2 @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_7930_sum_Oempty,axiom,
    ! [G2: real > rat] :
      ( ( groups1300246762558778688al_rat @ G2 @ bot_bot_set_real )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7931_sum_Oempty,axiom,
    ! [G2: real > nat] :
      ( ( groups1935376822645274424al_nat @ G2 @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_7932_sum_Oempty,axiom,
    ! [G2: real > int] :
      ( ( groups1932886352136224148al_int @ G2 @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty
thf(fact_7933_sum_Oempty,axiom,
    ! [G2: $o > real] :
      ( ( groups8691415230153176458o_real @ G2 @ bot_bot_set_o )
      = zero_zero_real ) ).

% sum.empty
thf(fact_7934_sum_Oempty,axiom,
    ! [G2: $o > rat] :
      ( ( groups7872700643590313910_o_rat @ G2 @ bot_bot_set_o )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7935_sum_Oempty,axiom,
    ! [G2: $o > nat] :
      ( ( groups8507830703676809646_o_nat @ G2 @ bot_bot_set_o )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_7936_sum_Oempty,axiom,
    ! [G2: $o > int] :
      ( ( groups8505340233167759370_o_int @ G2 @ bot_bot_set_o )
      = zero_zero_int ) ).

% sum.empty
thf(fact_7937_sum_Oempty,axiom,
    ! [G2: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G2 @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7938_sum_Oempty,axiom,
    ! [G2: nat > int] :
      ( ( groups3539618377306564664at_int @ G2 @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_7939_sum__eq__0__iff,axiom,
    ! [F2: set_int,F: int > nat] :
      ( ( finite_finite_int @ F2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X3: int] :
              ( ( member_int @ X3 @ F2 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7940_sum__eq__0__iff,axiom,
    ! [F2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X3: complex] :
              ( ( member_complex @ X3 @ F2 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7941_sum__eq__0__iff,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( ( groups977919841031483927at_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ F2 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7942_sum__eq__0__iff,axiom,
    ! [F2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( ( groups2027974829824023292at_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ F2 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7943_sum__eq__0__iff,axiom,
    ! [F2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ F2 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7944_sum_Oinfinite,axiom,
    ! [A2: set_int,G2: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G2 @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_7945_sum_Oinfinite,axiom,
    ! [A2: set_complex,G2: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G2 @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_7946_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > real] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G2 @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_7947_sum_Oinfinite,axiom,
    ! [A2: set_nat,G2: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G2 @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7948_sum_Oinfinite,axiom,
    ! [A2: set_int,G2: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G2 @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7949_sum_Oinfinite,axiom,
    ! [A2: set_complex,G2: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G2 @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7950_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > rat] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups1392844769737527556at_rat @ G2 @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7951_sum_Oinfinite,axiom,
    ! [A2: set_int,G2: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G2 @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_7952_sum_Oinfinite,axiom,
    ! [A2: set_complex,G2: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G2 @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_7953_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > nat] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2027974829824023292at_nat @ G2 @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_7954_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( real_V1803761363581548252l_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_7955_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( real_V4546457046886955230omplex @ X )
        = zero_zero_complex )
      = ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_7956_of__real__0,axiom,
    ( ( real_V1803761363581548252l_real @ zero_zero_real )
    = zero_zero_real ) ).

% of_real_0
thf(fact_7957_of__real__0,axiom,
    ( ( real_V4546457046886955230omplex @ zero_zero_real )
    = zero_zero_complex ) ).

% of_real_0
thf(fact_7958_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( real_V1803761363581548252l_real @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_7959_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( real_V4546457046886955230omplex @ X )
        = one_one_complex )
      = ( X = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_7960_of__real__1,axiom,
    ( ( real_V1803761363581548252l_real @ one_one_real )
    = one_one_real ) ).

% of_real_1
thf(fact_7961_of__real__1,axiom,
    ( ( real_V4546457046886955230omplex @ one_one_real )
    = one_one_complex ) ).

% of_real_1
thf(fact_7962_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_7963_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_7964_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_7965_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_7966_sum_Odelta,axiom,
    ! [S2: set_o,A: $o,B: $o > real] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7967_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7968_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7969_sum_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7970_sum_Odelta,axiom,
    ! [S2: set_o,A: $o,B: $o > rat] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7971_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7972_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7973_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7974_sum_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups1392844769737527556at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups1392844769737527556at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7975_sum_Odelta,axiom,
    ! [S2: set_o,A: $o,B: $o > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S2 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_7976_sum_Odelta_H,axiom,
    ! [S2: set_o,A: $o,B: $o > real] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7977_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7978_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7979_sum_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7980_sum_Odelta_H,axiom,
    ! [S2: set_o,A: $o,B: $o > rat] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7981_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7982_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7983_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7984_sum_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups1392844769737527556at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups1392844769737527556at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7985_sum_Odelta_H,axiom,
    ! [S2: set_o,A: $o,B: $o > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S2 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_7986_sum__abs,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_7987_sum__abs,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_7988_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G2: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups8097168146408367636l_real @ G2 @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_real @ ( G2 @ X ) @ ( groups8097168146408367636l_real @ G2 @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7989_sum_Oinsert,axiom,
    ! [A2: set_o,X: $o,G2: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( groups8691415230153176458o_real @ G2 @ ( insert_o @ X @ A2 ) )
          = ( plus_plus_real @ ( G2 @ X ) @ ( groups8691415230153176458o_real @ G2 @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7990_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G2: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups8778361861064173332t_real @ G2 @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_real @ ( G2 @ X ) @ ( groups8778361861064173332t_real @ G2 @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7991_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G2 @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_real @ ( G2 @ X ) @ ( groups5808333547571424918x_real @ G2 @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7992_sum_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ~ ( member_Extended_enat @ X @ A2 )
       => ( ( groups4148127829035722712t_real @ G2 @ ( insert_Extended_enat @ X @ A2 ) )
          = ( plus_plus_real @ ( G2 @ X ) @ ( groups4148127829035722712t_real @ G2 @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7993_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G2: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1300246762558778688al_rat @ G2 @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_rat @ ( G2 @ X ) @ ( groups1300246762558778688al_rat @ G2 @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7994_sum_Oinsert,axiom,
    ! [A2: set_o,X: $o,G2: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( groups7872700643590313910_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
          = ( plus_plus_rat @ ( G2 @ X ) @ ( groups7872700643590313910_o_rat @ G2 @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7995_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups2906978787729119204at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
          = ( plus_plus_rat @ ( G2 @ X ) @ ( groups2906978787729119204at_rat @ G2 @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7996_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G2: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_rat @ ( G2 @ X ) @ ( groups3906332499630173760nt_rat @ G2 @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7997_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G2 @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_rat @ ( G2 @ X ) @ ( groups5058264527183730370ex_rat @ G2 @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7998_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_7999_sum__abs__ge__zero,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_8000_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_8001_distinct__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
       => ( ( distinct_int @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
          = ( distinct_int @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_8002_distinct__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( distinct_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
          = ( distinct_VEBT_VEBT @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_8003_distinct__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( distinct_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_8004_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8005_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8006_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8007_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8008_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_8009_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_8010_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_8011_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ( size_s3451745648224563538omplex @ Xs2 )
                = N )
              & ( distinct_complex @ Xs2 )
              & ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8012_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs2: list_P6011104703257516679at_nat] :
              ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                = N )
              & ( distin6923225563576452346at_nat @ Xs2 )
              & ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8013_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs2: list_Extended_enat] :
              ( ( ( size_s3941691890525107288d_enat @ Xs2 )
                = N )
              & ( distin4523846830085650399d_enat @ Xs2 )
              & ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8014_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                = N )
              & ( distinct_VEBT_VEBT @ Xs2 )
              & ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8015_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = N )
              & ( distinct_nat @ Xs2 )
              & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8016_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ( size_size_list_int @ Xs2 )
                = N )
              & ( distinct_int @ Xs2 )
              & ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8017_norm__of__real__add1,axiom,
    ! [X: real] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ one_one_real ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8018_norm__of__real__add1,axiom,
    ! [X: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ one_one_complex ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8019_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
            @ A2 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_8020_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
            @ A2 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_8021_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
            @ A2 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_8022_sum__norm__le,axiom,
    ! [S2: set_o,F: $o > complex,G2: $o > real] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G2 @ X4 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5328290441151304332omplex @ F @ S2 ) ) @ ( groups8691415230153176458o_real @ G2 @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8023_sum__norm__le,axiom,
    ! [S2: set_set_nat,F: set_nat > complex,G2: set_nat > real] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G2 @ X4 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S2 ) ) @ ( groups5107569545109728110t_real @ G2 @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8024_sum__norm__le,axiom,
    ! [S2: set_set_nat_rat,F: set_nat_rat > complex,G2: set_nat_rat > real] :
      ( ! [X4: set_nat_rat] :
          ( ( member_set_nat_rat @ X4 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G2 @ X4 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6246630355582004071omplex @ F @ S2 ) ) @ ( groups4357547368389691109t_real @ G2 @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8025_sum__norm__le,axiom,
    ! [S2: set_int,F: int > complex,G2: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G2 @ X4 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G2 @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8026_sum__norm__le,axiom,
    ! [S2: set_nat,F: nat > complex,G2: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G2 @ X4 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G2 @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8027_sum__norm__le,axiom,
    ! [S2: set_complex,F: complex > complex,G2: complex > real] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G2 @ X4 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G2 @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8028_sum__norm__le,axiom,
    ! [S2: set_nat,F: nat > real,G2: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ S2 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X4 ) ) @ ( G2 @ X4 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G2 @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8029_sum_Oneutral,axiom,
    ! [A2: set_nat,G2: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( G2 @ X4 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G2 @ A2 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_8030_sum_Oneutral,axiom,
    ! [A2: set_complex,G2: complex > complex] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A2 )
         => ( ( G2 @ X4 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G2 @ A2 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_8031_sum_Oneutral,axiom,
    ! [A2: set_int,G2: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ( G2 @ X4 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G2 @ A2 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_8032_sum_Oneutral,axiom,
    ! [A2: set_nat,G2: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( G2 @ X4 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G2 @ A2 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_8033_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: $o > real,A2: set_o] :
      ( ( ( groups8691415230153176458o_real @ G2 @ A2 )
       != zero_zero_real )
     => ~ ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8034_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: int > real,A2: set_int] :
      ( ( ( groups8778361861064173332t_real @ G2 @ A2 )
       != zero_zero_real )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8035_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: $o > rat,A2: set_o] :
      ( ( ( groups7872700643590313910_o_rat @ G2 @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8036_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: nat > rat,A2: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G2 @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8037_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: int > rat,A2: set_int] :
      ( ( ( groups3906332499630173760nt_rat @ G2 @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8038_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: $o > nat,A2: set_o] :
      ( ( ( groups8507830703676809646_o_nat @ G2 @ A2 )
       != zero_zero_nat )
     => ~ ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8039_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: int > nat,A2: set_int] :
      ( ( ( groups4541462559716669496nt_nat @ G2 @ A2 )
       != zero_zero_nat )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8040_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: $o > int,A2: set_o] :
      ( ( ( groups8505340233167759370_o_int @ G2 @ A2 )
       != zero_zero_int )
     => ~ ! [A5: $o] :
            ( ( member_o @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8041_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: nat > int,A2: set_nat] :
      ( ( ( groups3539618377306564664at_int @ G2 @ A2 )
       != zero_zero_int )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8042_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: nat > nat,A2: set_nat] :
      ( ( ( groups3542108847815614940at_nat @ G2 @ A2 )
       != zero_zero_nat )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G2 @ A5 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8043_norm__sum,axiom,
    ! [F: nat > complex,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_8044_norm__sum,axiom,
    ! [F: complex > complex,A2: set_complex] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
      @ ( groups5808333547571424918x_real
        @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_8045_norm__sum,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_8046_choose__one,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ one_one_nat )
      = N ) ).

% choose_one
thf(fact_8047_sum__mono,axiom,
    ! [K4: set_o,F: $o > rat,G2: $o > rat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ K4 ) @ ( groups7872700643590313910_o_rat @ G2 @ K4 ) ) ) ).

% sum_mono
thf(fact_8048_sum__mono,axiom,
    ! [K4: set_nat,F: nat > rat,G2: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K4 ) @ ( groups2906978787729119204at_rat @ G2 @ K4 ) ) ) ).

% sum_mono
thf(fact_8049_sum__mono,axiom,
    ! [K4: set_int,F: int > rat,G2: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K4 ) @ ( groups3906332499630173760nt_rat @ G2 @ K4 ) ) ) ).

% sum_mono
thf(fact_8050_sum__mono,axiom,
    ! [K4: set_o,F: $o > nat,G2: $o > nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ K4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K4 ) @ ( groups8507830703676809646_o_nat @ G2 @ K4 ) ) ) ).

% sum_mono
thf(fact_8051_sum__mono,axiom,
    ! [K4: set_int,F: int > nat,G2: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K4 ) @ ( groups4541462559716669496nt_nat @ G2 @ K4 ) ) ) ).

% sum_mono
thf(fact_8052_sum__mono,axiom,
    ! [K4: set_o,F: $o > int,G2: $o > int] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ K4 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ K4 ) @ ( groups8505340233167759370_o_int @ G2 @ K4 ) ) ) ).

% sum_mono
thf(fact_8053_sum__mono,axiom,
    ! [K4: set_nat,F: nat > int,G2: nat > int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K4 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K4 ) @ ( groups3539618377306564664at_int @ G2 @ K4 ) ) ) ).

% sum_mono
thf(fact_8054_sum__mono,axiom,
    ! [K4: set_nat,F: nat > nat,G2: nat > nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K4 ) @ ( groups3542108847815614940at_nat @ G2 @ K4 ) ) ) ).

% sum_mono
thf(fact_8055_sum__mono,axiom,
    ! [K4: set_int,F: int > int,G2: int > int] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K4 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K4 ) @ ( groups4538972089207619220nt_int @ G2 @ K4 ) ) ) ).

% sum_mono
thf(fact_8056_sum__mono,axiom,
    ! [K4: set_nat,F: nat > real,G2: nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K4 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
     => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K4 ) @ ( groups6591440286371151544t_real @ G2 @ K4 ) ) ) ).

% sum_mono
thf(fact_8057_sum_Oswap__restrict,axiom,
    ! [A2: set_o,B2: set_nat,G2: $o > nat > nat,R: $o > nat > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups8507830703676809646_o_nat
            @ ^ [X3: $o] :
                ( groups3542108847815614940at_nat @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y2: nat] :
                ( groups8507830703676809646_o_nat
                @ ^ [X3: $o] : ( G2 @ X3 @ Y2 )
                @ ( collect_o
                  @ ^ [X3: $o] :
                      ( ( member_o @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8058_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G2: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X3: int] :
                ( groups3542108847815614940at_nat @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y2: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X3: int] : ( G2 @ X3 @ Y2 )
                @ ( collect_int
                  @ ^ [X3: int] :
                      ( ( member_int @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8059_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G2: complex > nat > nat,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups5693394587270226106ex_nat
            @ ^ [X3: complex] :
                ( groups3542108847815614940at_nat @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y2: nat] :
                ( groups5693394587270226106ex_nat
                @ ^ [X3: complex] : ( G2 @ X3 @ Y2 )
                @ ( collect_complex
                  @ ^ [X3: complex] :
                      ( ( member_complex @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8060_sum_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B2: set_nat,G2: extended_enat > nat > nat,R: extended_enat > nat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups2027974829824023292at_nat
            @ ^ [X3: extended_enat] :
                ( groups3542108847815614940at_nat @ ( G2 @ X3 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y2: nat] :
                ( groups2027974829824023292at_nat
                @ ^ [X3: extended_enat] : ( G2 @ X3 @ Y2 )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X3: extended_enat] :
                      ( ( member_Extended_enat @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8061_sum_Oswap__restrict,axiom,
    ! [A2: set_o,B2: set_complex,G2: $o > complex > complex,R: $o > complex > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5328290441151304332omplex
            @ ^ [X3: $o] :
                ( groups7754918857620584856omplex @ ( G2 @ X3 )
                @ ( collect_complex
                  @ ^ [Y2: complex] :
                      ( ( member_complex @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y2: complex] :
                ( groups5328290441151304332omplex
                @ ^ [X3: $o] : ( G2 @ X3 @ Y2 )
                @ ( collect_o
                  @ ^ [X3: $o] :
                      ( ( member_o @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8062_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_complex,G2: nat > complex > complex,R: nat > complex > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [X3: nat] :
                ( groups7754918857620584856omplex @ ( G2 @ X3 )
                @ ( collect_complex
                  @ ^ [Y2: complex] :
                      ( ( member_complex @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y2: complex] :
                ( groups2073611262835488442omplex
                @ ^ [X3: nat] : ( G2 @ X3 @ Y2 )
                @ ( collect_nat
                  @ ^ [X3: nat] :
                      ( ( member_nat @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8063_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_complex,G2: int > complex > complex,R: int > complex > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups3049146728041665814omplex
            @ ^ [X3: int] :
                ( groups7754918857620584856omplex @ ( G2 @ X3 )
                @ ( collect_complex
                  @ ^ [Y2: complex] :
                      ( ( member_complex @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y2: complex] :
                ( groups3049146728041665814omplex
                @ ^ [X3: int] : ( G2 @ X3 @ Y2 )
                @ ( collect_int
                  @ ^ [X3: int] :
                      ( ( member_int @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8064_sum_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B2: set_complex,G2: extended_enat > complex > complex,R: extended_enat > complex > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups6818542070133387226omplex
            @ ^ [X3: extended_enat] :
                ( groups7754918857620584856omplex @ ( G2 @ X3 )
                @ ( collect_complex
                  @ ^ [Y2: complex] :
                      ( ( member_complex @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y2: complex] :
                ( groups6818542070133387226omplex
                @ ^ [X3: extended_enat] : ( G2 @ X3 @ Y2 )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X3: extended_enat] :
                      ( ( member_Extended_enat @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8065_sum_Oswap__restrict,axiom,
    ! [A2: set_o,B2: set_int,G2: $o > int > int,R: $o > int > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups8505340233167759370_o_int
            @ ^ [X3: $o] :
                ( groups4538972089207619220nt_int @ ( G2 @ X3 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y2: int] :
                ( groups8505340233167759370_o_int
                @ ^ [X3: $o] : ( G2 @ X3 @ Y2 )
                @ ( collect_o
                  @ ^ [X3: $o] :
                      ( ( member_o @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8066_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_int,G2: nat > int > int,R: nat > int > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [X3: nat] :
                ( groups4538972089207619220nt_int @ ( G2 @ X3 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ A2 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y2: int] :
                ( groups3539618377306564664at_int
                @ ^ [X3: nat] : ( G2 @ X3 @ Y2 )
                @ ( collect_nat
                  @ ^ [X3: nat] :
                      ( ( member_nat @ X3 @ A2 )
                      & ( R @ X3 @ Y2 ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8067_sum__nonpos,axiom,
    ! [A2: set_o,F: $o > real] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8068_sum__nonpos,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8069_sum__nonpos,axiom,
    ! [A2: set_o,F: $o > rat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8070_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8071_sum__nonpos,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8072_sum__nonpos,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8073_sum__nonpos,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8074_sum__nonpos,axiom,
    ! [A2: set_o,F: $o > int] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
     => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ A2 ) @ zero_zero_int ) ) ).

% sum_nonpos
thf(fact_8075_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).

% sum_nonpos
thf(fact_8076_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8077_sum__nonneg,axiom,
    ! [A2: set_o,F: $o > real] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8078_sum__nonneg,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8079_sum__nonneg,axiom,
    ! [A2: set_o,F: $o > rat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8080_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8081_sum__nonneg,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8082_sum__nonneg,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8083_sum__nonneg,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8084_sum__nonneg,axiom,
    ! [A2: set_o,F: $o > int] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups8505340233167759370_o_int @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8085_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8086_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8087_sum__mono__inv,axiom,
    ! [F: $o > rat,I5: set_o,G2: $o > rat,I: $o] :
      ( ( ( groups7872700643590313910_o_rat @ F @ I5 )
        = ( groups7872700643590313910_o_rat @ G2 @ I5 ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
       => ( ( member_o @ I @ I5 )
         => ( ( finite_finite_o @ I5 )
           => ( ( F @ I )
              = ( G2 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8088_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G2: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G2 @ I5 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G2 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8089_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G2: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G2 @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G2 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8090_sum__mono__inv,axiom,
    ! [F: complex > rat,I5: set_complex,G2: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
        = ( groups5058264527183730370ex_rat @ G2 @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G2 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8091_sum__mono__inv,axiom,
    ! [F: extended_enat > rat,I5: set_Extended_enat,G2: extended_enat > rat,I: extended_enat] :
      ( ( ( groups1392844769737527556at_rat @ F @ I5 )
        = ( groups1392844769737527556at_rat @ G2 @ I5 ) )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
       => ( ( member_Extended_enat @ I @ I5 )
         => ( ( finite4001608067531595151d_enat @ I5 )
           => ( ( F @ I )
              = ( G2 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8092_sum__mono__inv,axiom,
    ! [F: $o > nat,I5: set_o,G2: $o > nat,I: $o] :
      ( ( ( groups8507830703676809646_o_nat @ F @ I5 )
        = ( groups8507830703676809646_o_nat @ G2 @ I5 ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
       => ( ( member_o @ I @ I5 )
         => ( ( finite_finite_o @ I5 )
           => ( ( F @ I )
              = ( G2 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8093_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G2: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G2 @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G2 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8094_sum__mono__inv,axiom,
    ! [F: complex > nat,I5: set_complex,G2: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
        = ( groups5693394587270226106ex_nat @ G2 @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G2 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8095_sum__mono__inv,axiom,
    ! [F: extended_enat > nat,I5: set_Extended_enat,G2: extended_enat > nat,I: extended_enat] :
      ( ( ( groups2027974829824023292at_nat @ F @ I5 )
        = ( groups2027974829824023292at_nat @ G2 @ I5 ) )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
       => ( ( member_Extended_enat @ I @ I5 )
         => ( ( finite4001608067531595151d_enat @ I5 )
           => ( ( F @ I )
              = ( G2 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8096_sum__mono__inv,axiom,
    ! [F: $o > int,I5: set_o,G2: $o > int,I: $o] :
      ( ( ( groups8505340233167759370_o_int @ F @ I5 )
        = ( groups8505340233167759370_o_int @ G2 @ I5 ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I5 )
           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
       => ( ( member_o @ I @ I5 )
         => ( ( finite_finite_o @ I5 )
           => ( ( F @ I )
              = ( G2 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8097_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_8098_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_8099_finite__distinct__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( ( set_VEBT_VEBT2 @ Xs3 )
            = A2 )
          & ( distinct_VEBT_VEBT @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8100_finite__distinct__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( ( set_nat2 @ Xs3 )
            = A2 )
          & ( distinct_nat @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8101_finite__distinct__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( ( set_int2 @ Xs3 )
            = A2 )
          & ( distinct_int @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8102_finite__distinct__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( ( set_complex2 @ Xs3 )
            = A2 )
          & ( distinct_complex @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8103_finite__distinct__list,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ? [Xs3: list_P6011104703257516679at_nat] :
          ( ( ( set_Pr5648618587558075414at_nat @ Xs3 )
            = A2 )
          & ( distin6923225563576452346at_nat @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8104_finite__distinct__list,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ? [Xs3: list_Extended_enat] :
          ( ( ( set_Extended_enat2 @ Xs3 )
            = A2 )
          & ( distin4523846830085650399d_enat @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8105_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_8106_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > nat,G2: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ ( suc @ X4 ) @ A2 )
           => ( ( F @ ( suc @ X4 ) )
              = ( G2 @ ( suc @ X4 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( groups3542108847815614940at_nat @ G2 @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_8107_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > real,G2: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ ( suc @ X4 ) @ A2 )
           => ( ( F @ ( suc @ X4 ) )
              = ( G2 @ ( suc @ X4 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A2 )
          = ( groups6591440286371151544t_real @ G2 @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_8108_sum_Ointer__filter,axiom,
    ! [A2: set_o,G2: $o > real,P: $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups8691415230153176458o_real @ G2
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups8691415230153176458o_real
          @ ^ [X3: $o] : ( if_real @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8109_sum_Ointer__filter,axiom,
    ! [A2: set_int,G2: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G2
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8110_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G2: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G2
          @ ( collect_complex
            @ ^ [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8111_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > real,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G2
          @ ( collec4429806609662206161d_enat
            @ ^ [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups4148127829035722712t_real
          @ ^ [X3: extended_enat] : ( if_real @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8112_sum_Ointer__filter,axiom,
    ! [A2: set_o,G2: $o > rat,P: $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups7872700643590313910_o_rat @ G2
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups7872700643590313910_o_rat
          @ ^ [X3: $o] : ( if_rat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8113_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G2: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G2
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8114_sum_Ointer__filter,axiom,
    ! [A2: set_int,G2: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G2
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X3: int] : ( if_rat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8115_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G2: complex > rat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G2
          @ ( collect_complex
            @ ^ [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X3: complex] : ( if_rat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8116_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > rat,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups1392844769737527556at_rat @ G2
          @ ( collec4429806609662206161d_enat
            @ ^ [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups1392844769737527556at_rat
          @ ^ [X3: extended_enat] : ( if_rat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8117_sum_Ointer__filter,axiom,
    ! [A2: set_o,G2: $o > nat,P: $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups8507830703676809646_o_nat @ G2
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups8507830703676809646_o_nat
          @ ^ [X3: $o] : ( if_nat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8118_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: nat > nat,M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_8119_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: nat > real,M2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_8120_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: nat > nat,M2: nat,K: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G2 @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_8121_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: nat > real,M2: nat,K: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G2 @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_8122_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_o,F: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X3: $o] :
                ( ( member_o @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8123_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X3: int] :
                ( ( member_int @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8124_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8125_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8126_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_o,F: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X3: $o] :
                ( ( member_o @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8127_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8128_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X3: int] :
                ( ( member_int @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8129_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8130_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups1392844769737527556at_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8131_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
       => ( ( ( groups8507830703676809646_o_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X3: $o] :
                ( ( member_o @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8132_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G2: int > real,I: int > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8133_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G2: complex > real,I: complex > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8134_sum__le__included,axiom,
    ! [S: set_int,T: set_Extended_enat,G2: extended_enat > real,I: extended_enat > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups4148127829035722712t_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8135_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G2: int > real,I: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8136_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G2: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8137_sum__le__included,axiom,
    ! [S: set_complex,T: set_Extended_enat,G2: extended_enat > real,I: extended_enat > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups4148127829035722712t_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8138_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_int,G2: int > real,I: int > extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8139_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_complex,G2: complex > real,I: complex > extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8140_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_Extended_enat,G2: extended_enat > real,I: extended_enat > extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S ) @ ( groups4148127829035722712t_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8141_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G2: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G2 @ X4 ) ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8142_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > real,G2: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
       => ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8143_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > real,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
       => ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8144_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A2 )
           => ( ord_less_eq_real @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
       => ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
         => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( groups4148127829035722712t_real @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8145_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > rat,G2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
       => ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8146_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > rat,G2: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
       => ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8147_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > rat,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
       => ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8148_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > rat,G2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
       => ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
         => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A2 ) @ ( groups1392844769737527556at_rat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8149_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > nat,G2: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
       => ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8150_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > nat,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
       => ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8151_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat,G2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
       => ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
         => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8152_sum_Orelated,axiom,
    ! [R: real > real > $o,S2: set_int,H: int > real,G2: int > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X24: real,Y24: real] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups8778361861064173332t_real @ H @ S2 ) @ ( groups8778361861064173332t_real @ G2 @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8153_sum_Orelated,axiom,
    ! [R: real > real > $o,S2: set_complex,H: complex > real,G2: complex > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X24: real,Y24: real] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups5808333547571424918x_real @ H @ S2 ) @ ( groups5808333547571424918x_real @ G2 @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8154_sum_Orelated,axiom,
    ! [R: real > real > $o,S2: set_Extended_enat,H: extended_enat > real,G2: extended_enat > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X24: real,Y24: real] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X24 @ Y24 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups4148127829035722712t_real @ H @ S2 ) @ ( groups4148127829035722712t_real @ G2 @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8155_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S2: set_nat,H: nat > rat,G2: nat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X24: rat,Y24: rat] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X24 @ Y24 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups2906978787729119204at_rat @ H @ S2 ) @ ( groups2906978787729119204at_rat @ G2 @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8156_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S2: set_int,H: int > rat,G2: int > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X24: rat,Y24: rat] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups3906332499630173760nt_rat @ H @ S2 ) @ ( groups3906332499630173760nt_rat @ G2 @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8157_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S2: set_complex,H: complex > rat,G2: complex > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X24: rat,Y24: rat] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups5058264527183730370ex_rat @ H @ S2 ) @ ( groups5058264527183730370ex_rat @ G2 @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8158_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S2: set_Extended_enat,H: extended_enat > rat,G2: extended_enat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X24: rat,Y24: rat] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X24 @ Y24 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups1392844769737527556at_rat @ H @ S2 ) @ ( groups1392844769737527556at_rat @ G2 @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8159_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S2: set_int,H: int > nat,G2: int > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X24: nat,Y24: nat] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X24 @ Y24 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups4541462559716669496nt_nat @ H @ S2 ) @ ( groups4541462559716669496nt_nat @ G2 @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8160_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S2: set_complex,H: complex > nat,G2: complex > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X24: nat,Y24: nat] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X24 @ Y24 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups5693394587270226106ex_nat @ H @ S2 ) @ ( groups5693394587270226106ex_nat @ G2 @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8161_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S2: set_Extended_enat,H: extended_enat > nat,G2: extended_enat > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X24: nat,Y24: nat] :
            ( ( ( R @ X1 @ X24 )
              & ( R @ Y1 @ Y24 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X24 @ Y24 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S2 )
               => ( R @ ( H @ X4 ) @ ( G2 @ X4 ) ) )
           => ( R @ ( groups2027974829824023292at_nat @ H @ S2 ) @ ( groups2027974829824023292at_nat @ G2 @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8162_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > real,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8163_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
         => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( groups4148127829035722712t_real @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8164_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > real,G2: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ A2 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8165_sum__strict__mono,axiom,
    ! [A2: set_o,F: $o > real,G2: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ A2 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
         => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A2 ) @ ( groups8691415230153176458o_real @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8166_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > real,G2: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8167_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > rat,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8168_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > rat,G2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
         => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A2 ) @ ( groups1392844769737527556at_rat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8169_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > rat,G2: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ A2 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8170_sum__strict__mono,axiom,
    ! [A2: set_o,F: $o > rat,G2: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ A2 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
         => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A2 ) @ ( groups7872700643590313910_o_rat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8171_sum__strict__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G2 @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8172_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_8173_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G2: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G2 @ ( insert_real @ X @ A2 ) )
            = ( groups8097168146408367636l_real @ G2 @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G2 @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_real @ ( G2 @ X ) @ ( groups8097168146408367636l_real @ G2 @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8174_sum_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G2: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( groups8691415230153176458o_real @ G2 @ ( insert_o @ X @ A2 ) )
            = ( groups8691415230153176458o_real @ G2 @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( groups8691415230153176458o_real @ G2 @ ( insert_o @ X @ A2 ) )
            = ( plus_plus_real @ ( G2 @ X ) @ ( groups8691415230153176458o_real @ G2 @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8175_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G2: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G2 @ ( insert_int @ X @ A2 ) )
            = ( groups8778361861064173332t_real @ G2 @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G2 @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_real @ ( G2 @ X ) @ ( groups8778361861064173332t_real @ G2 @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8176_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G2 @ ( insert_complex @ X @ A2 ) )
            = ( groups5808333547571424918x_real @ G2 @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G2 @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_real @ ( G2 @ X ) @ ( groups5808333547571424918x_real @ G2 @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8177_sum_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ X @ A2 )
         => ( ( groups4148127829035722712t_real @ G2 @ ( insert_Extended_enat @ X @ A2 ) )
            = ( groups4148127829035722712t_real @ G2 @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ X @ A2 )
         => ( ( groups4148127829035722712t_real @ G2 @ ( insert_Extended_enat @ X @ A2 ) )
            = ( plus_plus_real @ ( G2 @ X ) @ ( groups4148127829035722712t_real @ G2 @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8178_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G2: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G2 @ ( insert_real @ X @ A2 ) )
            = ( groups1300246762558778688al_rat @ G2 @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G2 @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_rat @ ( G2 @ X ) @ ( groups1300246762558778688al_rat @ G2 @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8179_sum_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G2: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( groups7872700643590313910_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
            = ( groups7872700643590313910_o_rat @ G2 @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( groups7872700643590313910_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
            = ( plus_plus_rat @ ( G2 @ X ) @ ( groups7872700643590313910_o_rat @ G2 @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8180_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
            = ( groups2906978787729119204at_rat @ G2 @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
            = ( plus_plus_rat @ ( G2 @ X ) @ ( groups2906978787729119204at_rat @ G2 @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8181_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G2: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
            = ( groups3906332499630173760nt_rat @ G2 @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_rat @ ( G2 @ X ) @ ( groups3906332499630173760nt_rat @ G2 @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8182_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G2 @ ( insert_complex @ X @ A2 ) )
            = ( groups5058264527183730370ex_rat @ G2 @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G2 @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_rat @ ( G2 @ X ) @ ( groups5058264527183730370ex_rat @ G2 @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8183_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_o,S2: set_o,I: $o > $o,J: $o > $o,T3: set_o,G2: $o > real,H: $o > real] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [A5: $o] :
              ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
               => ( member_o @ ( J @ A5 ) @ ( minus_minus_set_o @ T3 @ T5 ) ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: $o] :
                    ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                   => ( member_o @ ( I @ B5 ) @ ( minus_minus_set_o @ S2 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: $o] :
                        ( ( member_o @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: $o] :
                          ( ( member_o @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups8691415230153176458o_real @ G2 @ S2 )
                        = ( groups8691415230153176458o_real @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8184_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_int,S2: set_o,I: int > $o,J: $o > int,T3: set_int,G2: $o > real,H: int > real] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A5: $o] :
              ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_o @ ( I @ B5 ) @ ( minus_minus_set_o @ S2 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: $o] :
                          ( ( member_o @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups8691415230153176458o_real @ G2 @ S2 )
                        = ( groups8778361861064173332t_real @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8185_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_complex,S2: set_o,I: complex > $o,J: $o > complex,T3: set_complex,G2: $o > real,H: complex > real] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A5: $o] :
              ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
               => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: complex] :
                    ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_o @ ( I @ B5 ) @ ( minus_minus_set_o @ S2 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: complex] :
                        ( ( member_complex @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: $o] :
                          ( ( member_o @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups8691415230153176458o_real @ G2 @ S2 )
                        = ( groups5808333547571424918x_real @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8186_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_Extended_enat,S2: set_o,I: extended_enat > $o,J: $o > extended_enat,T3: set_Extended_enat,G2: $o > real,H: extended_enat > real] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite4001608067531595151d_enat @ T5 )
       => ( ! [A5: $o] :
              ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o @ A5 @ ( minus_minus_set_o @ S2 @ S5 ) )
               => ( member_Extended_enat @ ( J @ A5 ) @ ( minus_925952699566721837d_enat @ T3 @ T5 ) ) )
           => ( ! [B5: extended_enat] :
                  ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: extended_enat] :
                    ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                   => ( member_o @ ( I @ B5 ) @ ( minus_minus_set_o @ S2 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: extended_enat] :
                        ( ( member_Extended_enat @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: $o] :
                          ( ( member_o @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups8691415230153176458o_real @ G2 @ S2 )
                        = ( groups4148127829035722712t_real @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8187_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_o,S2: set_int,I: $o > int,J: int > $o,T3: set_o,G2: int > real,H: $o > real] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_o @ ( J @ A5 ) @ ( minus_minus_set_o @ T3 @ T5 ) ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: $o] :
                    ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: $o] :
                        ( ( member_o @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups8778361861064173332t_real @ G2 @ S2 )
                        = ( groups8691415230153176458o_real @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8188_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G2: int > real,H: int > real] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups8778361861064173332t_real @ G2 @ S2 )
                        = ( groups8778361861064173332t_real @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8189_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G2: int > real,H: complex > real] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: complex] :
                    ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: complex] :
                        ( ( member_complex @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups8778361861064173332t_real @ G2 @ S2 )
                        = ( groups5808333547571424918x_real @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8190_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_Extended_enat,S2: set_int,I: extended_enat > int,J: int > extended_enat,T3: set_Extended_enat,G2: int > real,H: extended_enat > real] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite4001608067531595151d_enat @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_Extended_enat @ ( J @ A5 ) @ ( minus_925952699566721837d_enat @ T3 @ T5 ) ) )
           => ( ! [B5: extended_enat] :
                  ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: extended_enat] :
                    ( ( member_Extended_enat @ B5 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: extended_enat] :
                        ( ( member_Extended_enat @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups8778361861064173332t_real @ G2 @ S2 )
                        = ( groups4148127829035722712t_real @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8191_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_o,S2: set_complex,I: $o > complex,J: complex > $o,T3: set_o,G2: complex > real,H: $o > real] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [A5: complex] :
              ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_o @ ( J @ A5 ) @ ( minus_minus_set_o @ T3 @ T5 ) ) )
           => ( ! [B5: $o] :
                  ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: $o] :
                    ( ( member_o @ B5 @ ( minus_minus_set_o @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B5 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A5: complex] :
                      ( ( member_complex @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: $o] :
                        ( ( member_o @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: complex] :
                          ( ( member_complex @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups5808333547571424918x_real @ G2 @ S2 )
                        = ( groups8691415230153176458o_real @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8192_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_int,S2: set_complex,I: int > complex,J: complex > int,T3: set_int,G2: complex > real,H: int > real] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A5: complex] :
              ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B5 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A5: complex] :
                      ( ( member_complex @ A5 @ S5 )
                     => ( ( G2 @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: complex] :
                          ( ( member_complex @ A5 @ S2 )
                         => ( ( H @ ( J @ A5 ) )
                            = ( G2 @ A5 ) ) )
                     => ( ( groups5808333547571424918x_real @ G2 @ S2 )
                        = ( groups8778361861064173332t_real @ H @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8193_choose__mult,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M2 ) @ ( binomial @ M2 @ K ) )
          = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M2 @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_8194_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_8195_distinct__conv__nth,axiom,
    ( distinct_int
    = ( ^ [Xs2: list_int] :
        ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs2 ) )
             => ( ( I4 != J3 )
               => ( ( nth_int @ Xs2 @ I4 )
                 != ( nth_int @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_8196_distinct__conv__nth,axiom,
    ( distinct_VEBT_VEBT
    = ( ^ [Xs2: list_VEBT_VEBT] :
        ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( ( I4 != J3 )
               => ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
                 != ( nth_VEBT_VEBT @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_8197_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs2: list_nat] :
        ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( I4 != J3 )
               => ( ( nth_nat @ Xs2 @ I4 )
                 != ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_8198_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( ( distinct_int @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ( ( nth_int @ Xs @ I )
              = ( nth_int @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_8199_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,J: nat] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( ( ( nth_VEBT_VEBT @ Xs @ I )
              = ( nth_VEBT_VEBT @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_8200_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ( ( nth_nat @ Xs @ I )
              = ( nth_nat @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_8201_sum__nonneg__0,axiom,
    ! [S: set_o,F: $o > real,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ S )
            = zero_zero_real )
         => ( ( member_o @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8202_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8203_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8204_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: extended_enat > real,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_Extended_enat @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8205_sum__nonneg__0,axiom,
    ! [S: set_o,F: $o > rat,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_o @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8206_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8207_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8208_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8209_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: extended_enat > rat,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1392844769737527556at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_Extended_enat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8210_sum__nonneg__0,axiom,
    ! [S: set_o,F: $o > nat,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups8507830703676809646_o_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_o @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8211_sum__nonneg__leq__bound,axiom,
    ! [S: set_o,F: $o > real,B2: real,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ S )
            = B2 )
         => ( ( member_o @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8212_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B2: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B2 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8213_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B2: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B2 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8214_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: extended_enat > real,B2: real,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ S )
            = B2 )
         => ( ( member_Extended_enat @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8215_sum__nonneg__leq__bound,axiom,
    ! [S: set_o,F: $o > rat,B2: rat,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ S )
            = B2 )
         => ( ( member_o @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8216_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B2: rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B2 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8217_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B2: rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B2 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8218_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B2: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B2 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8219_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: extended_enat > rat,B2: rat,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1392844769737527556at_rat @ F @ S )
            = B2 )
         => ( ( member_Extended_enat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8220_sum__nonneg__leq__bound,axiom,
    ! [S: set_o,F: $o > nat,B2: nat,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups8507830703676809646_o_nat @ F @ S )
            = B2 )
         => ( ( member_o @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8221_n__subsets,axiom,
    ! [A2: set_nat_rat,K: nat] :
      ( ( finite7830837933032798814at_rat @ A2 )
     => ( ( finite8736671560171388117at_rat
          @ ( collect_set_nat_rat
            @ ^ [B6: set_nat_rat] :
                ( ( ord_le2679597024174929757at_rat @ B6 @ A2 )
                & ( ( finite_card_nat_rat @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_nat_rat @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8222_n__subsets,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite2364142230527598318st_nat
          @ ( collect_set_list_nat
            @ ^ [B6: set_list_nat] :
                ( ( ord_le6045566169113846134st_nat @ B6 @ A2 )
                & ( ( finite_card_list_nat @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_list_nat @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8223_n__subsets,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite1149291290879098388et_nat
          @ ( collect_set_set_nat
            @ ^ [B6: set_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ B6 @ A2 )
                & ( ( finite_card_set_nat @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_set_nat @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8224_n__subsets,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [B6: set_nat] :
                ( ( ord_less_eq_set_nat @ B6 @ A2 )
                & ( ( finite_card_nat @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_nat @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8225_n__subsets,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite903997441450111292omplex
          @ ( collect_set_complex
            @ ^ [B6: set_complex] :
                ( ( ord_le211207098394363844omplex @ B6 @ A2 )
                & ( ( finite_card_complex @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_complex @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8226_n__subsets,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite4356350796350151305at_nat
          @ ( collec5514110066124741708at_nat
            @ ^ [B6: set_Pr1261947904930325089at_nat] :
                ( ( ord_le3146513528884898305at_nat @ B6 @ A2 )
                & ( ( finite711546835091564841at_nat @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite711546835091564841at_nat @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8227_n__subsets,axiom,
    ! [A2: set_Extended_enat,K: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite3719263829065406702d_enat
          @ ( collec2260605976452661553d_enat
            @ ^ [B6: set_Extended_enat] :
                ( ( ord_le7203529160286727270d_enat @ B6 @ A2 )
                & ( ( finite121521170596916366d_enat @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite121521170596916366d_enat @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8228_n__subsets,axiom,
    ! [A2: set_int,K: nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_card_set_int
          @ ( collect_set_int
            @ ^ [B6: set_int] :
                ( ( ord_less_eq_set_int @ B6 @ A2 )
                & ( ( finite_card_int @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_int @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8229_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G2: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G2
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G2 @ X3 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G2 @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8230_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G2
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G2 @ X3 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G2 @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8231_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G2
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X3: extended_enat] :
                  ( ( G2 @ X3 )
                  = zero_zero_real ) ) ) )
        = ( groups4148127829035722712t_real @ G2 @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8232_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G2: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G2
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G2 @ X3 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G2 @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8233_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G2
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G2 @ X3 )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G2 @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8234_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups1392844769737527556at_rat @ G2
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X3: extended_enat] :
                  ( ( G2 @ X3 )
                  = zero_zero_rat ) ) ) )
        = ( groups1392844769737527556at_rat @ G2 @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8235_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G2: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G2
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G2 @ X3 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G2 @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8236_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G2
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G2 @ X3 )
                  = zero_zero_nat ) ) ) )
        = ( groups5693394587270226106ex_nat @ G2 @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8237_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2027974829824023292at_nat @ G2
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X3: extended_enat] :
                  ( ( G2 @ X3 )
                  = zero_zero_nat ) ) ) )
        = ( groups2027974829824023292at_nat @ G2 @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8238_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G2: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G2
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G2 @ X3 )
                  = zero_zero_int ) ) ) )
        = ( groups5690904116761175830ex_int @ G2 @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8239_sum__power__add,axiom,
    ! [X: complex,M2: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I4: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M2 @ I4 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8240_sum__power__add,axiom,
    ! [X: rat,M2: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M2 @ I4 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8241_sum__power__add,axiom,
    ! [X: int,M2: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M2 @ I4 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8242_sum__power__add,axiom,
    ! [X: real,M2: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M2 @ I4 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8243_exp__sum,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( exp_real @ ( groups8778361861064173332t_real @ F @ I5 ) )
        = ( groups2316167850115554303t_real
          @ ^ [X3: int] : ( exp_real @ ( F @ X3 ) )
          @ I5 ) ) ) ).

% exp_sum
thf(fact_8244_exp__sum,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( exp_real @ ( groups5808333547571424918x_real @ F @ I5 ) )
        = ( groups766887009212190081x_real
          @ ^ [X3: complex] : ( exp_real @ ( F @ X3 ) )
          @ I5 ) ) ) ).

% exp_sum
thf(fact_8245_exp__sum,axiom,
    ! [I5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real] :
      ( ( finite6177210948735845034at_nat @ I5 )
     => ( ( exp_real @ ( groups4567486121110086003t_real @ F @ I5 ) )
        = ( groups6036352826371341000t_real
          @ ^ [X3: product_prod_nat_nat] : ( exp_real @ ( F @ X3 ) )
          @ I5 ) ) ) ).

% exp_sum
thf(fact_8246_exp__sum,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( exp_real @ ( groups4148127829035722712t_real @ F @ I5 ) )
        = ( groups97031904164794029t_real
          @ ^ [X3: extended_enat] : ( exp_real @ ( F @ X3 ) )
          @ I5 ) ) ) ).

% exp_sum
thf(fact_8247_exp__sum,axiom,
    ! [I5: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( exp_complex @ ( groups7754918857620584856omplex @ F @ I5 ) )
        = ( groups3708469109370488835omplex
          @ ^ [X3: complex] : ( exp_complex @ ( F @ X3 ) )
          @ I5 ) ) ) ).

% exp_sum
thf(fact_8248_exp__sum,axiom,
    ! [I5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( exp_real @ ( groups6591440286371151544t_real @ F @ I5 ) )
        = ( groups129246275422532515t_real
          @ ^ [X3: nat] : ( exp_real @ ( F @ X3 ) )
          @ I5 ) ) ) ).

% exp_sum
thf(fact_8249_sum_OatLeastAtMost__rev,axiom,
    ! [G2: nat > nat,N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_8250_sum_OatLeastAtMost__rev,axiom,
    ! [G2: nat > real,N: nat,M2: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_8251_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_int ) )
       => ( ( suminf_int @ F )
          = ( groups3539618377306564664at_int @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_8252_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_nat ) )
       => ( ( suminf_nat @ F )
          = ( groups3542108847815614940at_nat @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_8253_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_real ) )
       => ( ( suminf_real @ F )
          = ( groups6591440286371151544t_real @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_8254_sum__pos2,axiom,
    ! [I5: set_o,I: $o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8255_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8256_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8257_sum__pos2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( member_Extended_enat @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8258_sum__pos2,axiom,
    ! [I5: set_o,I: $o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8259_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8260_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8261_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8262_sum__pos2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( member_Extended_enat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1392844769737527556at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8263_sum__pos2,axiom,
    ! [I5: set_o,I: $o,F: $o > nat] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8264_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8265_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8266_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8267_sum__pos,axiom,
    ! [I5: set_o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8268_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8269_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8270_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1392844769737527556at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8271_sum__pos,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8272_sum__pos,axiom,
    ! [I5: set_o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8273_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8274_norm__less__p1,axiom,
    ! [X: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X ) ) @ one_one_real ) ) ) ).

% norm_less_p1
thf(fact_8275_norm__less__p1,axiom,
    ! [X: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X ) ) @ one_one_complex ) ) ) ).

% norm_less_p1
thf(fact_8276_sum__bounded__above,axiom,
    ! [A2: set_o,F: $o > rat,K4: rat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A2 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8277_sum__bounded__above,axiom,
    ! [A2: set_complex,F: complex > rat,K4: rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A2 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A2 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8278_sum__bounded__above,axiom,
    ! [A2: set_nat,F: nat > rat,K4: rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A2 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8279_sum__bounded__above,axiom,
    ! [A2: set_int,F: int > rat,K4: rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A2 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8280_sum__bounded__above,axiom,
    ! [A2: set_o,F: $o > nat,K4: nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A2 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8281_sum__bounded__above,axiom,
    ! [A2: set_complex,F: complex > nat,K4: nat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A2 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_complex @ A2 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8282_sum__bounded__above,axiom,
    ! [A2: set_int,F: int > nat,K4: nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A2 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8283_sum__bounded__above,axiom,
    ! [A2: set_o,F: $o > int,K4: int] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_o @ A2 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8284_sum__bounded__above,axiom,
    ! [A2: set_complex,F: complex > int,K4: int] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A2 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_complex @ A2 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8285_sum__bounded__above,axiom,
    ! [A2: set_nat,F: nat > int,K4: int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A2 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8286_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_8287_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_8288_binomial__mono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K7 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K7 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K7 ) ) ) ) ).

% binomial_mono
thf(fact_8289_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_8290_binomial__maximum,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_8291_binomial__antimono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K7 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K7 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_8292_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_8293_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_8294_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_8295_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_8296_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_8297_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( ( ord_less_nat @ K @ K7 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K7 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_8298_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( ( ord_less_nat @ K @ K7 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K7 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K7 ) ) ) ) ).

% binomial_strict_mono
thf(fact_8299_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_8300_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_8301_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_8302_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( 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_8303_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_8304_Sum__Icc__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_8305_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_8306_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [K2: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L4 @ ( 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 @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L4 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_8307_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_8308_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X3: complex] : X3
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_8309_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X3: complex] : X3
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_8310_Sum__Icc__int,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ M2 @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X3: int] : X3
          @ ( set_or1266510415728281911st_int @ M2 @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M2 @ ( minus_minus_int @ M2 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_8311_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I2: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I2 @ J2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
             => ( P @ I2 @ J2 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_8312_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_8313_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_8314_card__lessThan,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
      = U ) ).

% card_lessThan
thf(fact_8315_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8316_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_8317_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_8318_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_8319_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_8320_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_8321_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_8322_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8323_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_8324_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_8325_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [K2: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).

% finite_nat_bounded
thf(fact_8326_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_8327_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ M2 ) ) ).

% choose_rising_sum(2)
thf(fact_8328_sum__choose__diagonal,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M2 @ K3 ) )
          @ ( set_ord_atMost_nat @ M2 ) )
        = ( binomial @ ( suc @ N ) @ M2 ) ) ) ).

% sum_choose_diagonal
thf(fact_8329_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_8330_polynomial__product__nat,axiom,
    ! [M2: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M2 @ I2 )
         => ( ( A @ I2 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X @ I4 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_8331_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_8332_binomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% binomial_r_part_sum
thf(fact_8333_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_8334_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D6: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D6 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D6 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_8335_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
            & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8336_nat__dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( ( dvd_dvd_nat @ M2 @ one_one_nat )
      = ( M2 = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_8337_dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( ( dvd_dvd_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = ( M2
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_8338_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_8339_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_8340_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_8341_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_8342_even__diff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).

% even_diff_nat
thf(fact_8343_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_8344_dvd__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ M2 @ N )
     => ( ( dvd_dvd_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% dvd_antisym
thf(fact_8345_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_8346_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_8347_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_8348_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_8349_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_8350_dvd__diff__nat,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M2 )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% dvd_diff_nat
thf(fact_8351_nat__dvd__not__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).

% nat_dvd_not_less
thf(fact_8352_dvd__pos__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M2 @ N )
       => ( ord_less_nat @ zero_zero_nat @ M2 ) ) ) ).

% dvd_pos_nat
thf(fact_8353_dvd__minus__self,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% dvd_minus_self
thf(fact_8354_less__eq__dvd__minus,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( dvd_dvd_nat @ M2 @ N )
        = ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_8355_dvd__diffD1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
     => ( ( dvd_dvd_nat @ K @ M2 )
       => ( ( ord_less_eq_nat @ N @ M2 )
         => ( dvd_dvd_nat @ K @ N ) ) ) ) ).

% dvd_diffD1
thf(fact_8356_dvd__diffD,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( ( ord_less_eq_nat @ N @ M2 )
         => ( dvd_dvd_nat @ K @ M2 ) ) ) ) ).

% dvd_diffD
thf(fact_8357_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_8358_dvd__mult__cancel,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_8359_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
        = ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_8360_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D6: nat,X4: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D6 @ A )
          & ( dvd_dvd_nat @ D6 @ B )
          & ( ( times_times_nat @ A @ X4 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D6 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_8361_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_8362_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M2: nat,Q4: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( ( modulo_modulo_nat @ M2 @ Q4 )
          = ( modulo_modulo_nat @ N @ Q4 ) )
        = ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_8363_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( ( dvd_dvd_nat @ D @ N )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_8364_sums__if_H,axiom,
    ! [G2: nat > real,X: real] :
      ( ( sums_real @ G2 @ X )
     => ( sums_real
        @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( G2 @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X ) ) ).

% sums_if'
thf(fact_8365_sums__if,axiom,
    ! [G2: nat > real,X: real,F: nat > real,Y: real] :
      ( ( sums_real @ G2 @ X )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( F @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G2 @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sums_if
thf(fact_8366_dvd__fact,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_nat @ M2 @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% dvd_fact
thf(fact_8367_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M2 ) ) ) ) ).

% finite_divisors_nat
thf(fact_8368_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N ) @ M2 )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_8369_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M2 ) @ M2 )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_8370_dvd__minus__add,axiom,
    ! [Q4: nat,N: nat,R2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ Q4 @ N )
     => ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M2 ) )
       => ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ Q4 ) )
          = ( dvd_dvd_nat @ M2 @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M2 ) @ Q4 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_8371_power__dvd__imp__le,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_8372_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ R2 @ N )
     => ( ( ord_less_eq_nat @ R2 @ M2 )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M2 @ R2 ) )
         => ( ( modulo_modulo_nat @ M2 @ N )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_8373_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_8374_dvd__power__iff__le,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M2 ) @ ( power_power_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% dvd_power_iff_le
thf(fact_8375_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).

% even_set_encode_iff
thf(fact_8376_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_8377_sum__split__even__odd,axiom,
    ! [F: nat > real,G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G2 @ I4 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( G2 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_split_even_odd
thf(fact_8378_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_8379_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8380_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_8381_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_8382_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_8383_sin__x__le__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).

% sin_x_le_x
thf(fact_8384_sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).

% sin_le_one
thf(fact_8385_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).

% abs_sin_x_le_abs_x
thf(fact_8386_zdvd__antisym__nonneg,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ( dvd_dvd_int @ M2 @ N )
         => ( ( dvd_dvd_int @ N @ M2 )
           => ( M2 = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_8387_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).

% sin_x_ge_neg_x
thf(fact_8388_sin__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_ge_zero
thf(fact_8389_sin__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).

% sin_ge_minus_one
thf(fact_8390_abs__sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_8391_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( ( dvd_dvd_int @ Z @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ Z @ N ) ) ) ).

% zdvd_imp_le
thf(fact_8392_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( ( dvd_dvd_int @ D @ I )
       => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).

% dvd_imp_le_int
thf(fact_8393_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( ( dvd_dvd_int @ D @ N )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_8394_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_8395_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_8396_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_8397_nat__dvd__iff,axiom,
    ! [Z: int,M2: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M2 ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_8398_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X )
                = ( sin_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_8399_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_8400_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_8401_sin__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_8402_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_8403_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_8404_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X4: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
            & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X4 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y4 )
                    = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% sin_total
thf(fact_8405_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_8406_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_8407_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_8408_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T6: real] :
          ( ( ord_less_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8409_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8410_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_8411_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_8412_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_8413_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T6 ) )
               => ( Y
                 != ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_8414_cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).

% cos_le_one
thf(fact_8415_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( arcosh_real @ ( cosh_real @ X ) )
        = X ) ) ).

% arcosh_cosh_real
thf(fact_8416_cosh__real__nonneg,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_nonneg
thf(fact_8417_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_8418_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_8419_cosh__real__ge__1,axiom,
    ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).

% cosh_real_ge_1
thf(fact_8420_sinh__le__cosh__real,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_le_cosh_real
thf(fact_8421_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ( cos_real @ X )
                = ( cos_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_8422_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_8423_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_8424_cos__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).

% cos_ge_minus_one
thf(fact_8425_abs__cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_8426_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_8427_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_8428_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_8429_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_8430_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_8431_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_8432_cos__is__zero,axiom,
    ? [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
      & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X4 )
        = zero_zero_real )
      & ! [Y4: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
            & ( ord_less_eq_real @ Y4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y4 )
              = zero_zero_real ) )
         => ( Y4 = X4 ) ) ) ).

% cos_is_zero
thf(fact_8433_cos__two__le__zero,axiom,
    ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_le_zero
thf(fact_8434_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_8435_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
            & ( ord_less_eq_real @ X4 @ pi )
            & ( ( cos_real @ X4 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ pi )
                  & ( ( cos_real @ Y4 )
                    = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% cos_total
thf(fact_8436_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
      & ( ord_less_eq_real @ Y3 @ pi )
      & ( ( sin_real @ Y3 )
        = ( sin_real @ X ) )
      & ( ( cos_real @ Y3 )
        = ( cos_real @ X ) ) ) ).

% sincos_principal_value
thf(fact_8437_sin__cos__le1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_8438_cos__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_ge_zero
thf(fact_8439_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ pi )
            & ( X
              = ( cos_real @ T6 ) )
            & ( Y
              = ( sin_real @ T6 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_8440_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_8441_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T6: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X
                = ( cos_real @ T6 ) )
              & ( Y
                = ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_8442_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T6 ) )
          & ( Y
            = ( sin_real @ T6 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_8443_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_8444_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T6: real] :
            ( ( ord_less_real @ X @ T6 )
            & ( ord_less_real @ T6 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_8445_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_8446_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_8447_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8448_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_8449_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
          & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X4 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_8450_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).

% tan_mono_le
thf(fact_8451_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_8452_arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin
thf(fact_8453_arcsin__pi,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin_pi
thf(fact_8454_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
              = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_8455_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
              = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_8456_sin__arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y ) )
          = Y ) ) ) ).

% sin_arcsin
thf(fact_8457_arcsin__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X ) )
          = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).

% arcsin_minus
thf(fact_8458_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_8459_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ( arcsin @ X )
            = ( arcsin @ Y ) )
          = ( X = Y ) ) ) ) ).

% arcsin_eq_iff
thf(fact_8460_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% arcsin_le_mono
thf(fact_8461_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_8462_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% arcsin_less_mono
thf(fact_8463_arcsin__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_8464_arcsin__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_8465_arcsin__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).

% arcsin_lbound
thf(fact_8466_arcsin__sin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X ) )
          = X ) ) ) ).

% arcsin_sin
thf(fact_8467_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X3: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
              & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X3 )
                = Y2 ) ) ) ) ) ).

% arcsin_def
thf(fact_8468_arcosh__real__def,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( arcosh_real @ X )
        = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_8469_num_Osize__gen_I2_J,axiom,
    ! [X23: num] :
      ( ( size_num @ ( bit0 @ X23 ) )
      = ( plus_plus_nat @ ( size_num @ X23 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_8470_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% real_sqrt_le_iff
thf(fact_8471_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_ge_0_iff
thf(fact_8472_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_8473_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ one_one_real @ Y ) ) ).

% real_sqrt_ge_1_iff
thf(fact_8474_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_8475_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X ) ) ).

% real_sqrt_pow2
thf(fact_8476_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% real_sqrt_pow2_iff
thf(fact_8477_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 ) ).

% take_bit_nat_less_eq_self
thf(fact_8478_take__bit__tightened__less__eq__nat,axiom,
    ! [M2: nat,N: nat,Q4: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M2 @ Q4 ) @ ( bit_se2925701944663578781it_nat @ N @ Q4 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_8479_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_le_mono
thf(fact_8480_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_8481_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_8482_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_zero
thf(fact_8483_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sqrt @ X )
          = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_8484_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_one
thf(fact_8485_take__bit__tightened__less__eq__int,axiom,
    ! [M2: nat,N: nat,K: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M2 @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_8486_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_8487_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_8488_real__div__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
        = ( sqrt @ X ) ) ) ).

% real_div_sqrt
thf(fact_8489_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_8490_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_8491_ln__neg__is__const,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ln_ln_real @ X )
        = ( the_real
          @ ^ [X3: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_8492_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
        = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_8493_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M2 )
        = M2 )
      = ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_8494_take__bit__nat__less__exp,axiom,
    ! [N: nat,M2: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_8495_take__bit__nat__eq__self,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M2 )
        = M2 ) ) ).

% take_bit_nat_eq_self
thf(fact_8496_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).

% real_le_rsqrt
thf(fact_8497_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
     => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_8498_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_8499_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 ) ) ).

% take_bit_nat_less_self_iff
thf(fact_8500_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_8501_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_8502_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_8503_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_8504_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_8505_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
     => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).

% sqrt_ge_absD
thf(fact_8506_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_8507_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_8508_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_less_lsqrt
thf(fact_8509_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_8510_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_8511_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_8512_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_8513_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_8514_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_8515_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( sqrt @ X ) @ N )
          = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_8516_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_8517_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_8518_powr__half__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X ) ) ) ).

% powr_half_sqrt
thf(fact_8519_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_8520_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_8521_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_8522_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_8523_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_8524_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_8525_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X3 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_8526_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X3 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_8527_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_8528_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
     => ( ( sin_real @ X )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_8529_cos__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_8530_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_8531_sin__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_8532_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N4: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N4 = zero_zero_nat )
            | ( ord_less_nat @ M3 @ N4 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q3: nat] : ( product_Pair_nat_nat @ ( suc @ Q3 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N4 ) @ N4 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_8533_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_8534_cos__arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y ) )
          = Y ) ) ) ).

% cos_arccos
thf(fact_8535_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_8536_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
     => ( ( ( arccos @ X )
          = ( arccos @ Y ) )
        = ( X = Y ) ) ) ).

% arccos_eq_iff
thf(fact_8537_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% arccos_le_mono
thf(fact_8538_arccos__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).

% arccos_lbound
thf(fact_8539_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_8540_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% arccos_less_mono
thf(fact_8541_arccos__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_8542_arccos__cos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( arccos @ ( cos_real @ X ) )
          = X ) ) ) ).

% arccos_cos
thf(fact_8543_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y ) )
        = Y ) ) ).

% cos_arccos_abs
thf(fact_8544_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
     => ( ( arccos @ ( cos_real @ Theta ) )
        = ( abs_abs_real @ Theta ) ) ) ).

% arccos_cos_eq_abs
thf(fact_8545_arccos__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_8546_arccos__cos2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( arccos @ ( cos_real @ X ) )
          = ( uminus_uminus_real @ X ) ) ) ) ).

% arccos_cos2
thf(fact_8547_arccos__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X ) )
          = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).

% arccos_minus
thf(fact_8548_arccos__def,axiom,
    ( arccos
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X3 )
              & ( ord_less_eq_real @ X3 @ pi )
              & ( ( cos_real @ X3 )
                = Y2 ) ) ) ) ) ).

% arccos_def
thf(fact_8549_arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
          & ( ( cos_real @ ( arccos @ Y ) )
            = Y ) ) ) ) ).

% arccos
thf(fact_8550_arccos__minus__abs,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X ) )
        = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).

% arccos_minus_abs
thf(fact_8551_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X3: real] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X3 )
              & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_8552_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_8553_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z2: int] :
                ( ( ord_less_eq_int @ D5 @ Z8 )
                & ( ord_less_int @ Z8 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_8554_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z2: int] :
                ( ( ord_less_eq_int @ D5 @ Z2 )
                & ( ord_less_int @ Z8 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_8555_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_8556_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_8557_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M2: 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 @ M2 ) ) @ ( 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 @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M2 @ N )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_8558_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M2: 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 @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) )
      & ( ~ ( ( ( 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 @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( 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 @ M2 ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_8559_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_8560_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_8561_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X3: rat] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 )
              & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_8562_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_8563_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_8564_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_rat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% less_eq_rat_def
thf(fact_8565_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R2 )
     => ~ ! [S3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S3 )
           => ! [T6: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T6 )
               => ( R2
                 != ( plus_plus_rat @ S3 @ T6 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_8566_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).

% AND_lower
thf(fact_8567_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).

% AND_upper1
thf(fact_8568_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_8569_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_8570_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_8571_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_8572_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_8573_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_8574_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_8575_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_8576_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_8577_semiring__norm_I73_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(73)
thf(fact_8578_semiring__norm_I72_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(72)
thf(fact_8579_semiring__norm_I70_J,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M2 ) @ one ) ).

% semiring_norm(70)
thf(fact_8580_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_8581_and__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_8582_semiring__norm_I79_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(79)
thf(fact_8583_semiring__norm_I74_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% semiring_norm(74)
thf(fact_8584_and__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_8585_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_8586_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_8587_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_8588_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_8589_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_8590_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_nat_numerals(2)
thf(fact_8591_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_nat_numerals(1)
thf(fact_8592_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X24: num] :
            ( Y
           != ( bit0 @ X24 ) )
       => ~ ! [X32: num] :
              ( Y
             != ( bit1 @ X32 ) ) ) ) ).

% num.exhaust
thf(fact_8593_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_8594_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_8595_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_8596_num_Osize__gen_I3_J,axiom,
    ! [X33: num] :
      ( ( size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_8597_num_Osize_I6_J,axiom,
    ! [X33: num] :
      ( ( size_size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_8598_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_8599_mod__exhaust__less__4,axiom,
    ! [M2: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = zero_zero_nat )
      | ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = one_one_nat )
      | ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      | ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_8600_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N4: nat] :
          ( if_nat
          @ ( ( M3 = zero_zero_nat )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_8601_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_8602_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_8603_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% tanh_real_le_iff
thf(fact_8604_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% tanh_real_nonneg_iff
thf(fact_8605_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_8606_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_8607_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_8608_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_8609_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K ) )
      = ( N
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_8610_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_8611_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_8612_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_8613_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_8614_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_8615_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_8616_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_8617_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_8618_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_8619_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_8620_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_8621_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).

% XOR_lower
thf(fact_8622_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_8623_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M3: nat,N4: nat] : ( bit_se6528837805403552850or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_8624_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_8625_lessThan__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_8626_atMost__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).

% atMost_nat_numeral
thf(fact_8627_XOR__upper,axiom,
    ! [X: int,N: nat,Y: 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 @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_8628_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_8629_Sum__Ico__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_8630_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_8631_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_8632_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).

% finite_atLeastLessThan
thf(fact_8633_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_8634_atLeastLessThan__singleton,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
      = ( insert_nat @ M2 @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_8635_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_8636_or__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(2)
thf(fact_8637_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_8638_or__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(1)
thf(fact_8639_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M3: nat,N4: nat] : ( bit_se1412395901928357646or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_8640_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_8641_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less_eq
thf(fact_8642_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_8643_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_8644_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_8645_atLeastLessThan0,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_8646_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_8647_Least__Suc2,axiom,
    ! [P: nat > $o,N: nat,Q: nat > $o,M2: nat] :
      ( ( P @ N )
     => ( ( Q @ M2 )
       => ( ~ ( P @ zero_zero_nat )
         => ( ! [K2: nat] :
                ( ( P @ ( suc @ K2 ) )
                = ( Q @ K2 ) )
           => ( ( ord_Least_nat @ P )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_8648_Least__Suc,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M3: nat] : ( P @ ( suc @ M3 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_8649_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_8650_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_8651_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_8652_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
     => ( A2
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_8653_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_8654_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_8655_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_8656_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_8657_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ S2 ) ) ).

% card_sum_le_nat_sum
thf(fact_8658_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M2 @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_8659_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_8660_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_8661_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_8662_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_8663_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_8664_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_8665_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_8666_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: nat > nat,B: nat > nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I2 @ J2 )
         => ( ( ord_less_nat @ J2 @ N )
           => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ N )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_8667_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size_gen(1)
thf(fact_8668_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size(3)
thf(fact_8669_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_8670_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_8671_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).

% card_atLeastLessThan_int
thf(fact_8672_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).

% OR_lower
thf(fact_8673_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_8674_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_8675_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_8676_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_8677_OR__upper,axiom,
    ! [X: int,N: nat,Y: 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 @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_8678_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_8679_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_8680_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_8681_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_8682_binomial__def,axiom,
    ( binomial
    = ( ^ [N4: nat,K3: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K5: set_nat] :
                ( ( member_set_nat @ K5 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
                & ( ( finite_card_nat @ K5 )
                  = K3 ) ) ) ) ) ) ).

% binomial_def
thf(fact_8683_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [R4: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R4 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
          & ! [N6: nat] :
              ( ( ord_less_nat @ N6 @ ( finite_card_nat @ S2 ) )
             => ( member_nat @ ( R4 @ N6 ) @ S2 ) ) ) ) ).

% finite_enumerate
thf(fact_8684_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
      = ( case_nat_o @ $true
        @ ^ [Uu3: nat] : $false
        @ Nat ) ) ).

% nat.disc_eq_case(1)
thf(fact_8685_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
      = ( case_nat_o @ $false
        @ ^ [Uu3: nat] : $true
        @ Nat ) ) ).

% nat.disc_eq_case(2)
thf(fact_8686_less__eq__nat_Osimps_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M2 ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_8687_max__Suc2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_max_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
        @ M2 ) ) ).

% max_Suc2
thf(fact_8688_max__Suc1,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M2 )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
        @ M2 ) ) ).

% max_Suc1
thf(fact_8689_diff__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M2 @ N ) ) ) ).

% diff_Suc
thf(fact_8690_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_8691_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X25: nat] : X25 ) ) ).

% pred_def
thf(fact_8692_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_8693_Inf__nat__def1,axiom,
    ! [K4: set_nat] :
      ( ( K4 != bot_bot_set_nat )
     => ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).

% Inf_nat_def1
thf(fact_8694_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_8695_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_8696_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_8697_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_8698_nat__of__integer__code__post_I1_J,axiom,
    ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_nat ) ).

% nat_of_integer_code_post(1)
thf(fact_8699_nat__of__integer__code__post_I2_J,axiom,
    ( ( code_nat_of_integer @ one_one_Code_integer )
    = one_one_nat ) ).

% nat_of_integer_code_post(2)
thf(fact_8700_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_8701_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_8702_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_8703_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_8704_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_8705_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_8706_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_8707_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_8708_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_8709_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_8710_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_8711_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_8712_bit__push__bit__iff__int,axiom,
    ! [M2: nat,K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M2 @ K ) @ N )
      = ( ( ord_less_eq_nat @ M2 @ N )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_8713_bit__push__bit__iff__nat,axiom,
    ! [M2: nat,Q4: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M2 @ Q4 ) @ N )
      = ( ( ord_less_eq_nat @ M2 @ N )
        & ( bit_se1148574629649215175it_nat @ Q4 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_8714_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M2: nat,K: int] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( bit_se1146084159140164899it_int @ K @ N )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M2 @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_8715_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M: nat] :
              ( ( ord_less_eq_nat @ N2 @ M )
             => ( ( bit_se1146084159140164899it_int @ K @ M )
                = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).

% int_bit_bound
thf(fact_8716_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa2 )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_8717_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
     => ( ( ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_8718_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X3 )
    @ ^ [X3: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X3 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_8719_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I @ J ) @ K )
        = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_8720_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_8721_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_8722_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_8723_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_8724_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_8725_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_8726_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa2 )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa2 )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_8727_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_8728_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_8729_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_8730_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_8731_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_8732_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_8733_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_8734_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_8735_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_8736_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_8737_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_8738_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_8739_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_8740_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_8741_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_8742_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_8743_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_8744_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_8745_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_8746_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_8747_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_8748_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_8749_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_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_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_8752_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_8753_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X8: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X8 ) ) ) ) ) ).

% Inf_real_def
thf(fact_8754_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S6: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_8755_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S6: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_8756_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_8757_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_8758_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X3: int] : ( plus_plus_int @ X3 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_8759_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_8760_suminf__eq__SUP__real,axiom,
    ! [X5: nat > real] :
      ( ( summable_real @ X5 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X5 @ I2 ) )
       => ( ( suminf_real @ X5 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X5 @ ( set_ord_lessThan_nat @ I4 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_8761_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_8762_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_8763_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_8764_UN__atMost__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atMost_UNIV
thf(fact_8765_range__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat )
        = S2 ) ) ).

% range_enumerate
thf(fact_8766_UNIV__nat__eq,axiom,
    ( top_top_set_nat
    = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).

% UNIV_nat_eq
thf(fact_8767_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_8768_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_8769_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_8770_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_8771_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_8772_root__def,axiom,
    ( root
    = ( ^ [N4: nat,X3: real] :
          ( if_real @ ( N4 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N4 ) )
            @ X3 ) ) ) ) ).

% root_def
thf(fact_8773_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_8774_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_8775_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_8776_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G2: real > real,G3: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
         => ( has_fi5821293074295781190e_real @ G2 @ ( G3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X4 ) ) )
       => ( ( ord_less_eq_real @ A @ B )
         => ( ord_less_eq_real @ ( G2 @ A ) @ ( G2 @ B ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_8777_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_8778_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_8779_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_8780_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_8781_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F6: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ( has_fi5821293074295781190e_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
       => ? [Z3: real] :
            ( ( ord_less_real @ A @ Z3 )
            & ( ord_less_real @ Z3 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F6 @ Z3 ) ) ) ) ) ) ).

% MVT2
thf(fact_8782_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_8783_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_8784_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X3: real] : ( power_power_real @ X3 @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S ) ) ).

% DERIV_pow
thf(fact_8785_DERIV__fun__pow,axiom,
    ! [G2: real > real,M2: real,X: real,N: nat] :
      ( ( has_fi5821293074295781190e_real @ G2 @ M2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X3: real] : ( power_power_real @ ( G2 @ X3 ) @ N )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G2 @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M2 )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_8786_DERIV__series_H,axiom,
    ! [F: real > nat > real,F6: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
      ( ! [N2: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X3: real] : ( F @ X3 @ N2 )
          @ ( F6 @ X0 @ N2 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
           => ( summable_real @ ( F @ X4 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ( summable_real @ ( F6 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N2: nat,X4: real,Y3: real] :
                    ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X4 @ N2 ) @ ( F @ Y3 @ N2 ) ) ) @ ( times_times_real @ ( L5 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y3 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X3: real] : ( suminf_real @ ( F @ X3 ) )
                  @ ( suminf_real @ ( F6 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_8787_DERIV__fun__powr,axiom,
    ! [G2: real > real,M2: real,X: real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G2 @ M2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G2 @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X3: real] : ( powr_real @ ( G2 @ X3 ) @ R2 )
          @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G2 @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M2 )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_8788_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_8789_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_8790_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_8791_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_8792_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T6: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
                  & ( ord_less_eq_real @ T6 @ H ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ T6 )
                & ( ord_less_real @ T6 @ H )
                & ( ( F @ H )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_8793_Maclaurin2,axiom,
    ! [H: real,Diff: nat > real > real,F: real > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T6 )
                & ( ord_less_eq_real @ T6 @ H ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ? [T6: real] :
              ( ( ord_less_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H )
              & ( ( F @ H )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_8794_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T6: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ H @ T6 )
                  & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ H @ T6 )
                & ( ord_less_real @ T6 @ zero_zero_real )
                & ( ( F @ H )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_8795_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( X != zero_zero_real )
         => ( ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
                & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_8796_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,T6: real] :
            ( ( ( ord_less_nat @ M4 @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_8797_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ A @ T6 )
                  & ( ord_less_real @ T6 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_8798_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ C @ T6 )
                  & ( ord_less_real @ T6 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_8799_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B )
                 => ( ( X != C )
                   => ? [T6: real] :
                        ( ( ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ X @ T6 )
                            & ( ord_less_real @ T6 @ C ) ) )
                        & ( ~ ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ C @ T6 )
                            & ( ord_less_real @ T6 @ X ) ) )
                        & ( ( F @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M3 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_8800_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: nat > real > real,K: nat,B2: real] :
      ( ! [M4: nat,T6: real] :
          ( ( ( ord_less_nat @ M4 @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ H ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K ) )
       => ! [M: nat,T7: real] :
            ( ( ( ord_less_nat @ M @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ H ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M ) ) )
                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M ) @ T7 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T7 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M ) ) ) )
                  @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_8801_summable__Leibniz_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
         => ! [N6: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_8802_summable__Leibniz_I2_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
         => ! [N6: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_8803_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N4: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_8804_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_8805_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_8806_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).

% mult_nat_left_at_top
thf(fact_8807_monoseq__convergent,axiom,
    ! [X5: nat > real,B2: real] :
      ( ( topolo6980174941875973593q_real @ X5 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X5 @ I2 ) ) @ B2 )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_8808_nested__sequence__unique,axiom,
    ! [F: nat > real,G2: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( G2 @ ( suc @ N2 ) ) @ ( G2 @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G2 @ N2 ) )
         => ( ( filterlim_nat_real
              @ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G2 @ N4 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N6: nat] : ( ord_less_eq_real @ L4 @ ( G2 @ N6 ) )
                & ( filterlim_nat_real @ G2 @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_8809_LIMSEQ__inverse__zero,axiom,
    ! [X5: nat > real] :
      ( ! [R4: real] :
        ? [N8: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N8 @ N2 )
         => ( ord_less_real @ R4 @ ( X5 @ N2 ) ) )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( inverse_inverse_real @ ( X5 @ N4 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_8810_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
       => ( ! [E: real] :
              ( ( ord_less_real @ zero_zero_real @ E )
             => ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_8811_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_8812_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( summable_real
            @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ) ).

% summable
thf(fact_8813_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_8814_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N4: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_8815_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_8816_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N6: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N4: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N6: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N4: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_8817_summable__Leibniz_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N4: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_8818_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_8819_Bseq__eq__bounded,axiom,
    ! [F: nat > real,A: real,B: real] :
      ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
     => ( bfun_nat_real @ F @ at_top_nat ) ) ).

% Bseq_eq_bounded
thf(fact_8820_Bseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_8821_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ B @ X4 )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
              & ( ord_less_real @ Y4 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_8822_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F2: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
            @ at_top_real
            @ F2 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_8823_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).

% at_top_le_at_infinity
thf(fact_8824_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).

% at_bot_le_at_infinity
thf(fact_8825_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X4: nat] :
          ( ( ord_less_eq_nat @ C @ X4 )
         => ( P @ X4 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_8826_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N3: nat] :
          ! [N4: nat] :
            ( ( ord_less_eq_nat @ N3 @ N4 )
           => ( P @ N4 ) ) ) ) ).

% eventually_sequentially
thf(fact_8827_le__sequentially,axiom,
    ! [F2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F2 @ at_top_nat )
      = ( ! [N3: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N3 ) @ F2 ) ) ) ).

% le_sequentially
thf(fact_8828_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ X4 @ B )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y4 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_8829_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F2: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
            @ at_bot_real
            @ F2 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_8830_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_8831_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_greaterThanAtMost
thf(fact_8832_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_8833_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_8834_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_8835_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_8836_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_8837_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_8838_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_8839_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).

% card_greaterThanAtMost_int
thf(fact_8840_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_8841_Gcd__eq__Max,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M5 )
         => ( ( gcd_Gcd_nat @ M5 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M3: nat] :
                      ( collect_nat
                      @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M3 ) )
                  @ M5 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_8842_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N ) ) )
        = N ) ) ).

% Max_divisors_self_nat
thf(fact_8843_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_8844_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).

% Sup_nat_def
thf(fact_8845_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N4: nat] :
          ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N4 ) @ M3 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_8846_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_8847_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_8848_INT__greaterThan__UNIV,axiom,
    ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
    = bot_bot_set_nat ) ).

% INT_greaterThan_UNIV
thf(fact_8849_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X3: nat] :
                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
                                       => ( ( ord_less_nat @ Mi3 @ X3 )
                                          & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_8850_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_8851_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_8852_decseq__bounded,axiom,
    ! [X5: nat > real,B2: real] :
      ( ( order_9091379641038594480t_real @ X5 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X5 @ I2 ) )
       => ( bfun_nat_real @ X5 @ at_top_nat ) ) ) ).

% decseq_bounded
thf(fact_8853_decseq__convergent,axiom,
    ! [X5: nat > real,B2: real] :
      ( ( order_9091379641038594480t_real @ X5 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X5 @ I2 ) )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I3: nat] : ( ord_less_eq_real @ L6 @ ( X5 @ I3 ) ) ) ) ) ).

% decseq_convergent
thf(fact_8854_UN__atLeast__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atLeast_UNIV
thf(fact_8855_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_8856_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
      = ( ( Deg = Deg4 )
        & ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
           => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
            & ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
               => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                    & ! [X3: nat] :
                        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_8857_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                   => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                    & ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I4: nat] :
                            ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                            & ! [X3: nat] :
                                ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
                                 => ( ( ord_less_nat @ Mi3 @ X3 )
                                    & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_8858_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                      & ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I4: nat] :
                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X3: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                               => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                              & ! [X3: nat] :
                                  ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
                                   => ( ( ord_less_nat @ Mi3 @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_8859_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X3: nat] :
                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
                                       => ( ( ord_less_nat @ Mi3 @ X3 )
                                          & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_8860_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X3: nat] :
                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
                                       => ( ( ord_less_nat @ Mi3 @ X3 )
                                          & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_8861_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                        & ! [X3: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I4: nat] :
                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X3: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                & ! [X3: nat] :
                                    ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
                                     => ( ( ord_less_nat @ Mi3 @ X3 )
                                        & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_8862_Sup__real__def,axiom,
    ( comple1385675409528146559p_real
    = ( ^ [X8: set_real] :
          ( ord_Least_real
          @ ^ [Z2: real] :
            ! [X3: real] :
              ( ( member_real @ X3 @ X8 )
             => ( ord_less_eq_real @ X3 @ Z2 ) ) ) ) ) ).

% Sup_real_def
thf(fact_8863_Sup__int__def,axiom,
    ( complete_Sup_Sup_int
    = ( ^ [X8: set_int] :
          ( the_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ X8 )
              & ! [Y2: int] :
                  ( ( member_int @ Y2 @ X8 )
                 => ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ) ) ).

% Sup_int_def
thf(fact_8864_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_8865_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_8866_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_8867_incseq__bounded,axiom,
    ! [X5: nat > real,B2: real] :
      ( ( order_mono_nat_real @ X5 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( X5 @ I2 ) @ B2 )
       => ( bfun_nat_real @ X5 @ at_top_nat ) ) ) ).

% incseq_bounded
thf(fact_8868_incseq__convergent,axiom,
    ! [X5: nat > real,B2: real] :
      ( ( order_mono_nat_real @ X5 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( X5 @ I2 ) @ B2 )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I3: nat] : ( ord_less_eq_real @ ( X5 @ I3 ) @ L6 ) ) ) ) ).

% incseq_convergent
thf(fact_8869_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S2 ) ) ) ) ).

% infinite_int_iff_infinite_nat_abs
thf(fact_8870_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_8871_take__bit__num__simps_I1_J,axiom,
    ! [M2: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M2 )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_8872_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: nat > real,G2: nat > nat] :
      ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
     => ( ( order_mono_nat_real @ F )
       => ( ( order_5726023648592871131at_nat @ G2 )
         => ( ( bfun_nat_real
              @ ^ [X3: nat] : ( F @ ( G2 @ X3 ) )
              @ at_top_nat )
            = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
thf(fact_8873_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( order_5726023648592871131at_nat @ F )
     => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).

% strict_mono_imp_increasing
thf(fact_8874_infinite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ? [R4: nat > nat] :
          ( ( order_5726023648592871131at_nat @ R4 )
          & ! [N6: nat] : ( member_nat @ ( R4 @ N6 ) @ S2 ) ) ) ).

% infinite_enumerate
thf(fact_8875_strict__mono__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( order_5726023648592871131at_nat @ ( infini8530281810654367211te_nat @ S2 ) ) ) ).

% strict_mono_enumerate
thf(fact_8876_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N4: nat,M3: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M3 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_8877_num__of__nat__numeral__eq,axiom,
    ! [Q4: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q4 ) )
      = Q4 ) ).

% num_of_nat_numeral_eq
thf(fact_8878_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_8879_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_8880_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_8881_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_8882_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( plus_plus_nat @ M2 @ N ) )
          = ( plus_plus_num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_8883_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_8884_min__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).

% min_Suc_Suc
thf(fact_8885_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_8886_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_8887_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_8888_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_8889_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_8890_min__diff,axiom,
    ! [M2: nat,I: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I ) @ ( minus_minus_nat @ N @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I ) ) ).

% min_diff
thf(fact_8891_nat__mult__min__right,axiom,
    ! [M2: nat,N: nat,Q4: nat] :
      ( ( times_times_nat @ M2 @ ( ord_min_nat @ N @ Q4 ) )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q4 ) ) ) ).

% nat_mult_min_right
thf(fact_8892_nat__mult__min__left,axiom,
    ! [M2: nat,N: nat,Q4: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M2 @ N ) @ Q4 )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).

% nat_mult_min_left
thf(fact_8893_min__Suc1,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M2 )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
        @ M2 ) ) ).

% min_Suc1
thf(fact_8894_min__Suc2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_min_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
        @ M2 ) ) ).

% min_Suc2
thf(fact_8895_inj__Suc,axiom,
    ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).

% inj_Suc
thf(fact_8896_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ N5 )
         => ( ord_less_eq_nat @ K @ N2 ) )
     => ( inj_on_nat_nat
        @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K )
        @ N5 ) ) ).

% inj_on_diff_nat
thf(fact_8897_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_8898_summable__reindex,axiom,
    ! [F: nat > real,G2: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G2 @ top_top_set_nat )
       => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
         => ( summable_real @ ( comp_nat_real_nat @ F @ G2 ) ) ) ) ) ).

% summable_reindex
thf(fact_8899_suminf__reindex__mono,axiom,
    ! [F: nat > real,G2: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G2 @ top_top_set_nat )
       => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
         => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G2 ) ) @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_8900_suminf__reindex,axiom,
    ! [F: nat > real,G2: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G2 @ top_top_set_nat )
       => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
         => ( ! [X4: nat] :
                ( ~ ( member_nat @ X4 @ ( image_nat_nat @ G2 @ top_top_set_nat ) )
               => ( ( F @ X4 )
                  = zero_zero_real ) )
           => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G2 ) )
              = ( suminf_real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_8901_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A2 @ F )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ A2 )
           => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
       => ( topolo5044208981011980120l_real @ A2
          @ ^ [X3: real] : ( arcosh_real @ ( F @ X3 ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_8902_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ? [C3: real,D6: real] :
            ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
              = ( set_or1222579329274155063t_real @ C3 @ D6 ) )
            & ( ord_less_eq_real @ C3 @ D6 ) ) ) ) ).

% continuous_image_closed_interval
thf(fact_8903_continuous__on__arcosh,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).

% continuous_on_arcosh
thf(fact_8904_continuous__on__artanh,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).

% continuous_on_artanh
thf(fact_8905_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: real > real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X4: real] :
              ( ( ord_less_real @ A @ X4 )
             => ( ( ord_less_real @ X4 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A @ X )
           => ( ( ord_less_eq_real @ X @ B )
             => ( ( F @ X )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_8906_powr__real__of__int_H,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( X != zero_zero_real )
          | ( ord_less_int @ zero_zero_int @ N ) )
       => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
          = ( power_int_real @ X @ N ) ) ) ) ).

% powr_real_of_int'
thf(fact_8907_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ( ord_less_eq_real @ A @ X4 )
              & ( ord_less_eq_real @ X4 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
       => ? [L6: real,M9: real] :
            ( ! [X2: real] :
                ( ( ( ord_less_eq_real @ A @ X2 )
                  & ( ord_less_eq_real @ X2 @ B ) )
               => ( ( ord_less_eq_real @ L6 @ ( F @ X2 ) )
                  & ( ord_less_eq_real @ ( F @ X2 ) @ M9 ) ) )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ L6 @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ M9 ) )
               => ? [X4: real] :
                    ( ( ord_less_eq_real @ A @ X4 )
                    & ( ord_less_eq_real @ X4 @ B )
                    & ( ( F @ X4 )
                      = Y4 ) ) ) ) ) ) ).

% isCont_Lb_Ub
thf(fact_8908_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G2: real > real,F: real > real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ A @ Z3 )
             => ( ( ord_less_eq_real @ Z3 @ B )
               => ( ( G2 @ ( F @ Z3 ) )
                  = Z3 ) ) )
         => ( ! [Z3: real] :
                ( ( ord_less_eq_real @ A @ Z3 )
               => ( ( ord_less_eq_real @ Z3 @ B )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G2 ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_8909_LIM__less__bound,axiom,
    ! [B: real,X: real,F: real > real] :
      ( ( ord_less_real @ B @ X )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).

% LIM_less_bound
thf(fact_8910_isCont__inverse__function,axiom,
    ! [D: real,X: real,G2: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
           => ( ( G2 @ ( F @ Z3 ) )
              = Z3 ) )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G2 ) ) ) ) ).

% isCont_inverse_function
thf(fact_8911_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G2: real > real,G3: real > real,F6: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z3: real] :
            ( ( ord_less_eq_real @ A @ Z3 )
           => ( ( ord_less_eq_real @ Z3 @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ A @ Z3 )
             => ( ( ord_less_eq_real @ Z3 @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G2 ) ) )
         => ( ! [Z3: real] :
                ( ( ord_less_real @ A @ Z3 )
               => ( ( ord_less_real @ Z3 @ B )
                 => ( has_fi5821293074295781190e_real @ G2 @ ( G3 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
           => ( ! [Z3: real] :
                  ( ( ord_less_real @ A @ Z3 )
                 => ( ( ord_less_real @ Z3 @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F6 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
             => ? [C3: real] :
                  ( ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G3 @ C3 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G2 @ B ) @ ( G2 @ A ) ) @ ( F6 @ C3 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_8912_bdd__above__nat,axiom,
    condit2214826472909112428ve_nat = finite_finite_nat ).

% bdd_above_nat
thf(fact_8913_GMVT,axiom,
    ! [A: real,B: real,F: real > real,G2: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ( ord_less_eq_real @ A @ X4 )
              & ( ord_less_eq_real @ X4 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
       => ( ! [X4: real] :
              ( ( ( ord_less_real @ A @ X4 )
                & ( ord_less_real @ X4 @ B ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
         => ( ! [X4: real] :
                ( ( ( ord_less_eq_real @ A @ X4 )
                  & ( ord_less_eq_real @ X4 @ B ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G2 ) )
           => ( ! [X4: real] :
                  ( ( ( ord_less_real @ A @ X4 )
                    & ( ord_less_real @ X4 @ B ) )
                 => ( differ6690327859849518006l_real @ G2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( ( has_fi5821293074295781190e_real @ G2 @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G2 @ B ) @ ( G2 @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_8914_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_8915_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_8916_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_8917_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X4: num] :
            ( ( P @ X4 )
           => ( P @ ( inc @ X4 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_8918_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_8919_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_8920_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_8921_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_8922_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_8923_BitM__inc__eq,axiom,
    ! [N: num] :
      ( ( bitM @ ( inc @ N ) )
      = ( bit1 @ N ) ) ).

% BitM_inc_eq
thf(fact_8924_mult__inc,axiom,
    ! [X: num,Y: num] :
      ( ( times_times_num @ X @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).

% mult_inc
thf(fact_8925_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I4: int,N4: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
          & ( N4 != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_8926_Rats__abs__iff,axiom,
    ! [X: real] :
      ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
      = ( member_real @ X @ field_5140801741446780682s_real ) ) ).

% Rats_abs_iff
thf(fact_8927_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X4: real] :
      ( ( member_real @ X4 @ field_5140801741446780682s_real )
      & ( ord_less_real @ X4 @ X ) ) ).

% Rats_no_bot_less
thf(fact_8928_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [X4: real] :
          ( ( member_real @ X4 @ field_5140801741446780682s_real )
          & ( ord_less_real @ X @ X4 )
          & ( ord_less_real @ X4 @ Y ) ) ) ).

% Rats_dense_in_real
thf(fact_8929_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X4: real] :
      ( ( member_real @ X4 @ field_5140801741446780682s_real )
      & ( ord_less_eq_real @ X @ X4 ) ) ).

% Rats_no_top_le
thf(fact_8930_Rats__eq__int__div__int,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I4: int,J3: int] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( ring_1_of_int_real @ J3 ) ) )
          & ( J3 != zero_zero_int ) ) ) ) ).

% Rats_eq_int_div_int
thf(fact_8931_Arg__bounded,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
      & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).

% Arg_bounded
thf(fact_8932_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( bij_betw_nat_complex
        @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        @ ( set_ord_lessThan_nat @ N )
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_8933_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ( arg @ Z )
            = X ) ) ) ) ).

% cis_Arg_unique
thf(fact_8934_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z )
          = ( cis @ ( arg @ Z ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).

% Arg_correct
thf(fact_8935_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_8936_bij__betw__Suc,axiom,
    ! [M5: set_nat,N5: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M5 @ N5 )
      = ( ( image_nat_nat @ suc @ M5 )
        = N5 ) ) ).

% bij_betw_Suc
thf(fact_8937_bij__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat @ S2 ) ) ).

% bij_enumerate
thf(fact_8938_Arg__def,axiom,
    ( arg
    = ( ^ [Z2: complex] :
          ( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A4: real] :
                ( ( ( sgn_sgn_complex @ Z2 )
                  = ( cis @ A4 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
                & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_8939_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y2: nat,Z2: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y2 @ V3 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
          @ ( rep_Integ @ X3 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_8940_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y2: nat,Z2: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y2 @ V3 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
          @ ( rep_Integ @ X3 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_8941_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
        @ Xa2
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_8942_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_8943_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N4: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N4 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_8944_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_8945_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
        @ Xa2
        @ X ) ) ).

% less_int.abs_eq
thf(fact_8946_card__length__sum__list__rec,axiom,
    ! [M2: nat,N5: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L3: list_nat] :
                ( ( ( size_size_list_nat @ L3 )
                  = M2 )
                & ( ( 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 @ M2 @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L3 )
                    = N5 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L3: list_nat] :
                  ( ( ( size_size_list_nat @ L3 )
                    = M2 )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L3 ) @ one_one_nat )
                    = N5 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_8947_card__length__sum__list,axiom,
    ! [M2: nat,N5: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L3: list_nat] :
              ( ( ( size_size_list_nat @ L3 )
                = M2 )
              & ( ( groups4561878855575611511st_nat @ L3 )
                = N5 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M2 ) @ one_one_nat ) @ N5 ) ) ).

% card_length_sum_list
thf(fact_8948_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_8949_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_8950_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_8951_take__upt,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M2 ) @ N )
     => ( ( take_nat @ M2 @ ( upt @ I @ N ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M2 ) ) ) ) ).

% take_upt
thf(fact_8952_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_8953_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_8954_sum__list__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M2 @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X3: nat] : X3
          @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ).

% sum_list_upt
thf(fact_8955_map__add__upt,axiom,
    ! [N: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% map_add_upt
thf(fact_8956_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).

% atLeast_upt
thf(fact_8957_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_8958_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M2 @ N ) ) ).

% sorted_wrt_upt
thf(fact_8959_sorted__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N ) ) ).

% sorted_upt
thf(fact_8960_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_8961_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N4 ) ) ) ) ) ).

% atMost_upto
thf(fact_8962_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_8963_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_8964_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_8965_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_8966_map__decr__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( upt @ M2 @ N ) ) ).

% map_decr_upt
thf(fact_8967_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_8968_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_8969_sorted__upto,axiom,
    ! [M2: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M2 @ N ) ) ).

% sorted_upto
thf(fact_8970_Field__natLeq__on,axiom,
    ! [N: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X3: nat,Y2: nat] :
                ( ( ord_less_nat @ X3 @ N )
                & ( ord_less_nat @ Y2 @ N )
                & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) )
      = ( collect_nat
        @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) ).

% Field_natLeq_on
thf(fact_8971_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_8972_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_8973_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P5: rat,Q3: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A4: int,C5: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D5: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D5 ) @ ( times_times_int @ C5 @ B4 ) )
              @ ( quotient_of @ Q3 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_8974_of__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).

% of_real_sqrt
thf(fact_8975_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( last_nat @ ( upt @ I @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_8976_sqr_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).

% sqr.simps(3)
thf(fact_8977_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_8978_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_8979_sqr_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).

% sqr.simps(2)
thf(fact_8980_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_8981_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X3: num] : ( times_times_num @ X3 @ X3 ) ) ) ).

% sqr_conv_mult
thf(fact_8982_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit1 @ Y ) )
      = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).

% pow.simps(3)
thf(fact_8983_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_8984_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit0 @ Y ) )
      = ( sqr @ ( pow @ X @ Y ) ) ) ).

% pow.simps(2)
thf(fact_8985_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_8986_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X3: nat,Y2: nat] : ( if_Pro3027730157355071871nt_int @ ( Y2 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y2 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_8987_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_8988_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_8989_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_8990_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_8991_finite__vimage__Suc__iff,axiom,
    ! [F2: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F2 ) )
      = ( finite_finite_nat @ F2 ) ) ).

% finite_vimage_Suc_iff
thf(fact_8992_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A2 ) )
      = ( vimage_nat_nat @ suc @ A2 ) ) ).

% vimage_Suc_insert_0
thf(fact_8993_natLeq__on__wo__rel,axiom,
    ! [N: nat] :
      ( bNF_We3818239936649020644el_nat
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ N )
              & ( ord_less_nat @ Y2 @ N )
              & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).

% natLeq_on_wo_rel
thf(fact_8994_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% pred_nat_trancl_eq_le
thf(fact_8995_pairs__le__eq__Sigma,axiom,
    ! [M2: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M2 ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M2 )
        @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M2 @ R5 ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_8996_less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% less_eq
thf(fact_8997_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: nat > real,M5: nat] :
      ( ( bfun_nat_real
        @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ M5 ) )
        @ at_top_nat )
     => ( ! [M4: nat,N2: nat] :
            ( ( ord_less_eq_nat @ M5 @ M4 )
           => ( ( ord_less_eq_nat @ M4 @ N2 )
             => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N2 ) ) ) )
       => ( topolo7531315842566124627t_real @ F ) ) ) ).

% Bseq_monoseq_convergent'_inc
thf(fact_8998_Bseq__mono__convergent,axiom,
    ! [X5: nat > real] :
      ( ( bfun_nat_real @ X5 @ at_top_nat )
     => ( ! [M4: nat,N2: nat] :
            ( ( ord_less_eq_nat @ M4 @ N2 )
           => ( ord_less_eq_real @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) )
       => ( topolo7531315842566124627t_real @ X5 ) ) ) ).

% Bseq_mono_convergent
thf(fact_8999_convergent__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( topolo7531315842566124627t_real @ ( power_power_real @ X ) ) ) ) ).

% convergent_realpow
thf(fact_9000_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: nat > real,M5: nat] :
      ( ( bfun_nat_real
        @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ M5 ) )
        @ at_top_nat )
     => ( ! [M4: nat,N2: nat] :
            ( ( ord_less_eq_nat @ M5 @ M4 )
           => ( ( ord_less_eq_nat @ M4 @ N2 )
             => ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ M4 ) ) ) )
       => ( topolo7531315842566124627t_real @ F ) ) ) ).

% Bseq_monoseq_convergent'_dec
thf(fact_9001_Restr__natLeq,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) )
          @ ^ [Uu3: nat] :
              ( collect_nat
              @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ N )
              & ( ord_less_nat @ Y2 @ N )
              & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_9002_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_9003_Restr__natLeq2,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
          @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ N )
              & ( ord_less_nat @ Y2 @ N )
              & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_9004_natLeq__underS__less,axiom,
    ! [N: nat] :
      ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
      = ( collect_nat
        @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) ).

% natLeq_underS_less
thf(fact_9005_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_9006_trans__pair__less,axiom,
    trans_4347625901269045472at_nat @ fun_pair_less ).

% trans_pair_less
thf(fact_9007_total__pair__less,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( total_3592101749530773125at_nat @ A2 @ fun_pair_less ) ).

% total_pair_less
thf(fact_9008_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z ) ) ).

% pair_less_iff1
thf(fact_9009_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat @ fun_pair_less ).

% wf_pair_less
thf(fact_9010_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_9011_gcd__nat_Oordering__top__axioms,axiom,
    ( ordering_top_nat @ dvd_dvd_nat
    @ ^ [M3: nat,N4: nat] :
        ( ( dvd_dvd_nat @ M3 @ N4 )
        & ( M3 != N4 ) )
    @ zero_zero_nat ) ).

% gcd_nat.ordering_top_axioms
thf(fact_9012_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X3 )
    @ ^ [X3: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X3 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_9013_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_9014_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_9015_pair__leq__def,axiom,
    ( fun_pair_leq
    = ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).

% pair_leq_def
thf(fact_9016_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ 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_9017_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_9018_wmin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_9019_wmax__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X5 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X5 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_9020_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_9021_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_9022_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y6 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y6 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X5 ) @ Y6 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_9023_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_9024_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y6 )
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y6 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_9025_smin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( X5 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_9026_min__strict__def,axiom,
    ( fun_min_strict
    = ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).

% min_strict_def
thf(fact_9027_max__strict__def,axiom,
    ( fun_max_strict
    = ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).

% max_strict_def
thf(fact_9028_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_9029_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_9030_bit__concat__bit__iff,axiom,
    ! [M2: nat,K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M2 @ K @ L ) @ N )
      = ( ( ( ord_less_nat @ N @ M2 )
          & ( bit_se1146084159140164899it_int @ K @ N ) )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9031_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_9032_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_9033_division__segment__nat__def,axiom,
    ( euclid3398187327856392827nt_nat
    = ( ^ [N4: nat] : one_one_nat ) ) ).

% division_segment_nat_def
thf(fact_9034_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_9035_less__eq__enat__def,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [M3: extended_enat] :
          ( extended_case_enat_o
          @ ^ [N1: nat] :
              ( extended_case_enat_o
              @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
              @ $false
              @ M3 )
          @ $true ) ) ) ).

% less_eq_enat_def
thf(fact_9036_less__enat__def,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [M3: extended_enat,N4: extended_enat] :
          ( extended_case_enat_o
          @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N4 )
          @ $false
          @ M3 ) ) ) ).

% less_enat_def
thf(fact_9037_of__rat__dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Q5: rat] :
          ( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q5 ) )
          & ( ord_less_real @ ( field_7254667332652039916t_real @ Q5 ) @ Y ) ) ) ).

% of_rat_dense
thf(fact_9038_compute__powr__real,axiom,
    ( powr_real2
    = ( ^ [B4: real,I4: real] :
          ( if_real @ ( ord_less_eq_real @ B4 @ zero_zero_real )
          @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            @ ^ [Uu3: product_unit] : ( powr_real2 @ B4 @ I4 ) )
          @ ( if_real
            @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I4 ) )
              = I4 )
            @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I4 ) @ ( power_power_real @ B4 @ ( nat2 @ ( archim6058952711729229775r_real @ I4 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B4 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I4 ) ) ) ) ) )
            @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ ^ [Uu3: product_unit] : ( powr_real2 @ B4 @ I4 ) ) ) ) ) ) ).

% compute_powr_real
thf(fact_9039_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_9040_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_9041_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_9042_char_Osize_I2_J,axiom,
    ! [X15: $o,X23: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X15 @ X23 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_9043_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_9044_char_Osize__gen,axiom,
    ! [X15: $o,X23: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_char @ ( char2 @ X15 @ X23 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size_gen
thf(fact_9045_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_9046_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( ( member_real @ X @ field_5140801741446780682s_real )
     => ~ ! [M4: nat,N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( ( ( abs_abs_real @ X )
                = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
             => ~ ( algebr934650988132801477me_nat @ M4 @ N2 ) ) ) ) ).

% Rats_abs_nat_div_natE
thf(fact_9047_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_9048_coprime__Suc__0__left,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).

% coprime_Suc_0_left
thf(fact_9049_coprime__Suc__0__right,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).

% coprime_Suc_0_right
thf(fact_9050_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_9051_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_9052_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_9053_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
      @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_9054_less__eq__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
      @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
    @ ord_less_eq_int
    @ ord_less_eq_int ) ).

% less_eq_integer.rsp
thf(fact_9055_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
      @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_9056_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_9057_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_9058_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
    @ pcr_int
    @ ^ [N4: nat] : ( product_Pair_nat_nat @ N4 @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_9059_Real_Opositive_Orsp,axiom,
    ( bNF_re728719798268516973at_o_o @ realrel
    @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
    @ ^ [X8: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N4: nat] :
            ( ( ord_less_eq_nat @ K3 @ N4 )
           => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) )
    @ ^ [X8: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N4: nat] :
            ( ( ord_less_eq_nat @ K3 @ N4 )
           => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) ) ) ).

% Real.positive.rsp
thf(fact_9060_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_9061_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_9062_transp__realrel,axiom,
    transp_nat_rat @ realrel ).

% transp_realrel
thf(fact_9063_one__real_Orsp,axiom,
    ( realrel
    @ ^ [N4: nat] : one_one_rat
    @ ^ [N4: nat] : one_one_rat ) ).

% one_real.rsp
thf(fact_9064_zero__real_Orsp,axiom,
    ( realrel
    @ ^ [N4: nat] : zero_zero_rat
    @ ^ [N4: nat] : zero_zero_rat ) ).

% zero_real.rsp
thf(fact_9065_uminus__real_Orsp,axiom,
    ( bNF_re895249473297799549at_rat @ realrel @ realrel
    @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) )
    @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) ) ) ).

% uminus_real.rsp
thf(fact_9066_times__real_Orsp,axiom,
    ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
    @ ^ [X8: nat > rat,Y7: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) )
    @ ^ [X8: nat > rat,Y7: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) ) ) ).

% times_real.rsp
thf(fact_9067_plus__real_Orsp,axiom,
    ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
    @ ^ [X8: nat > rat,Y7: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) )
    @ ^ [X8: nat > rat,Y7: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) ) ) ).

% plus_real.rsp
thf(fact_9068_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_9069_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).

% less_int.rsp
thf(fact_9070_inverse__real_Orsp,axiom,
    ( bNF_re895249473297799549at_rat @ realrel @ realrel
    @ ^ [X8: nat > rat] :
        ( if_nat_rat @ ( vanishes @ X8 )
        @ ^ [N4: nat] : zero_zero_rat
        @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) )
    @ ^ [X8: nat > rat] :
        ( if_nat_rat @ ( vanishes @ X8 )
        @ ^ [N4: nat] : zero_zero_rat
        @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) ) ) ).

% inverse_real.rsp
thf(fact_9071_vanishes__const,axiom,
    ! [C: rat] :
      ( ( vanishes
        @ ^ [N4: nat] : C )
      = ( C = zero_zero_rat ) ) ).

% vanishes_const
thf(fact_9072_vanishes__minus,axiom,
    ! [X5: nat > rat] :
      ( ( vanishes @ X5 )
     => ( vanishes
        @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X5 @ N4 ) ) ) ) ).

% vanishes_minus
thf(fact_9073_vanishes__diff,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( vanishes @ X5 )
     => ( ( vanishes @ Y6 )
       => ( vanishes
          @ ^ [N4: nat] : ( minus_minus_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ).

% vanishes_diff
thf(fact_9074_vanishes__add,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( vanishes @ X5 )
     => ( ( vanishes @ Y6 )
       => ( vanishes
          @ ^ [N4: nat] : ( plus_plus_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ).

% vanishes_add
thf(fact_9075_vanishesD,axiom,
    ! [X5: nat > rat,R2: rat] :
      ( ( vanishes @ X5 )
     => ( ( ord_less_rat @ zero_zero_rat @ R2 )
       => ? [K2: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ K2 @ N6 )
           => ( ord_less_rat @ ( abs_abs_rat @ ( X5 @ N6 ) ) @ R2 ) ) ) ) ).

% vanishesD
thf(fact_9076_vanishesI,axiom,
    ! [X5: nat > rat] :
      ( ! [R4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R4 )
         => ? [K8: nat] :
            ! [N2: nat] :
              ( ( ord_less_eq_nat @ K8 @ N2 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X5 @ N2 ) ) @ R4 ) ) )
     => ( vanishes @ X5 ) ) ).

% vanishesI
thf(fact_9077_vanishes__def,axiom,
    ( vanishes
    = ( ^ [X8: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K3: nat] :
            ! [N4: nat] :
              ( ( ord_less_eq_nat @ K3 @ N4 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N4 ) ) @ R5 ) ) ) ) ) ).

% vanishes_def
thf(fact_9078_vanishes__mult__bounded,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ? [A8: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ A8 )
          & ! [N2: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X5 @ N2 ) ) @ A8 ) )
     => ( ( vanishes @ Y6 )
       => ( vanishes
          @ ^ [N4: nat] : ( times_times_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ).

% vanishes_mult_bounded
thf(fact_9079_inverse__real_Otransfer,axiom,
    ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
    @ ^ [X8: nat > rat] :
        ( if_nat_rat @ ( vanishes @ X8 )
        @ ^ [N4: nat] : zero_zero_rat
        @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) )
    @ inverse_inverse_real ) ).

% inverse_real.transfer
thf(fact_9080_inverse__real_Oabs__eq,axiom,
    ! [X: nat > rat] :
      ( ( realrel @ X @ X )
     => ( ( inverse_inverse_real @ ( real2 @ X ) )
        = ( real2
          @ ( if_nat_rat @ ( vanishes @ X )
            @ ^ [N4: nat] : zero_zero_rat
            @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X @ N4 ) ) ) ) ) ) ).

% inverse_real.abs_eq
thf(fact_9081_real_Oabs__induct,axiom,
    ! [P: real > $o,X: real] :
      ( ! [Y3: nat > rat] :
          ( ( realrel @ Y3 @ Y3 )
         => ( P @ ( real2 @ Y3 ) ) )
     => ( P @ X ) ) ).

% real.abs_induct
thf(fact_9082_zero__real__def,axiom,
    ( zero_zero_real
    = ( real2
      @ ^ [N4: nat] : zero_zero_rat ) ) ).

% zero_real_def
thf(fact_9083_of__nat__Real,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [X3: nat] :
          ( real2
          @ ^ [N4: nat] : ( semiri681578069525770553at_rat @ X3 ) ) ) ) ).

% of_nat_Real
thf(fact_9084_one__real__def,axiom,
    ( one_one_real
    = ( real2
      @ ^ [N4: nat] : one_one_rat ) ) ).

% one_real_def
thf(fact_9085_of__rat__Real,axiom,
    ( field_7254667332652039916t_real
    = ( ^ [X3: rat] :
          ( real2
          @ ^ [N4: nat] : X3 ) ) ) ).

% of_rat_Real
thf(fact_9086_of__int__Real,axiom,
    ( ring_1_of_int_real
    = ( ^ [X3: int] :
          ( real2
          @ ^ [N4: nat] : ( ring_1_of_int_rat @ X3 ) ) ) ) ).

% of_int_Real
thf(fact_9087_real_Orel__eq__transfer,axiom,
    ( bNF_re4521903465945308077real_o @ pcr_real
    @ ( bNF_re4297313714947099218al_o_o @ pcr_real
      @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
    @ realrel
    @ ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) ) ).

% real.rel_eq_transfer
thf(fact_9088_zero__real_Otransfer,axiom,
    ( pcr_real
    @ ^ [N4: nat] : zero_zero_rat
    @ zero_zero_real ) ).

% zero_real.transfer
thf(fact_9089_one__real_Otransfer,axiom,
    ( pcr_real
    @ ^ [N4: nat] : one_one_rat
    @ one_one_real ) ).

% one_real.transfer
thf(fact_9090_uminus__real_Otransfer,axiom,
    ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
    @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) )
    @ uminus_uminus_real ) ).

% uminus_real.transfer
thf(fact_9091_plus__real_Otransfer,axiom,
    ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
    @ ^ [X8: nat > rat,Y7: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) )
    @ plus_plus_real ) ).

% plus_real.transfer
thf(fact_9092_uminus__real_Oabs__eq,axiom,
    ! [X: nat > rat] :
      ( ( realrel @ X @ X )
     => ( ( uminus_uminus_real @ ( real2 @ X ) )
        = ( real2
          @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X @ N4 ) ) ) ) ) ).

% uminus_real.abs_eq
thf(fact_9093_times__real_Otransfer,axiom,
    ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
    @ ^ [X8: nat > rat,Y7: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) )
    @ times_times_real ) ).

% times_real.transfer
thf(fact_9094_plus__real_Oabs__eq,axiom,
    ! [Xa2: nat > rat,X: nat > rat] :
      ( ( realrel @ Xa2 @ Xa2 )
     => ( ( realrel @ X @ X )
       => ( ( plus_plus_real @ ( real2 @ Xa2 ) @ ( real2 @ X ) )
          = ( real2
            @ ^ [N4: nat] : ( plus_plus_rat @ ( Xa2 @ N4 ) @ ( X @ N4 ) ) ) ) ) ) ).

% plus_real.abs_eq
thf(fact_9095_times__real_Oabs__eq,axiom,
    ! [Xa2: nat > rat,X: nat > rat] :
      ( ( realrel @ Xa2 @ Xa2 )
     => ( ( realrel @ X @ X )
       => ( ( times_times_real @ ( real2 @ Xa2 ) @ ( real2 @ X ) )
          = ( real2
            @ ^ [N4: nat] : ( times_times_rat @ ( Xa2 @ N4 ) @ ( X @ N4 ) ) ) ) ) ) ).

% times_real.abs_eq
thf(fact_9096_Real_Opositive_Otransfer,axiom,
    ( bNF_re4297313714947099218al_o_o @ pcr_real
    @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
    @ ^ [X8: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N4: nat] :
            ( ( ord_less_eq_nat @ K3 @ N4 )
           => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) )
    @ positive ) ).

% Real.positive.transfer
thf(fact_9097_Real_Opositive_Oabs__eq,axiom,
    ! [X: nat > rat] :
      ( ( realrel @ X @ X )
     => ( ( positive @ ( real2 @ X ) )
        = ( ? [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
              & ? [K3: nat] :
                ! [N4: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N4 )
                 => ( ord_less_rat @ R5 @ ( X @ N4 ) ) ) ) ) ) ) ).

% Real.positive.abs_eq
thf(fact_9098_Real_Opositive__mult,axiom,
    ! [X: real,Y: real] :
      ( ( positive @ X )
     => ( ( positive @ Y )
       => ( positive @ ( times_times_real @ X @ Y ) ) ) ) ).

% Real.positive_mult
thf(fact_9099_Real_Opositive__add,axiom,
    ! [X: real,Y: real] :
      ( ( positive @ X )
     => ( ( positive @ Y )
       => ( positive @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% Real.positive_add
thf(fact_9100_Real_Opositive__zero,axiom,
    ~ ( positive @ zero_zero_real ) ).

% Real.positive_zero
thf(fact_9101_Real_Opositive__minus,axiom,
    ! [X: real] :
      ( ~ ( positive @ X )
     => ( ( X != zero_zero_real )
       => ( positive @ ( uminus_uminus_real @ X ) ) ) ) ).

% Real.positive_minus
thf(fact_9102_less__real__def,axiom,
    ( ord_less_real
    = ( ^ [X3: real,Y2: real] : ( positive @ ( minus_minus_real @ Y2 @ X3 ) ) ) ) ).

% less_real_def
thf(fact_9103_le__Real,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( cauchy @ Y6 )
       => ( ( ord_less_eq_real @ ( real2 @ X5 ) @ ( real2 @ Y6 ) )
          = ( ! [R5: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ R5 )
               => ? [K3: nat] :
                  ! [N4: nat] :
                    ( ( ord_less_eq_nat @ K3 @ N4 )
                   => ( ord_less_eq_rat @ ( X5 @ N4 ) @ ( plus_plus_rat @ ( Y6 @ N4 ) @ R5 ) ) ) ) ) ) ) ) ).

% le_Real
thf(fact_9104_Real_Opositive_Orep__eq,axiom,
    ( positive
    = ( ^ [X3: real] :
        ? [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
          & ? [K3: nat] :
            ! [N4: nat] :
              ( ( ord_less_eq_nat @ K3 @ N4 )
             => ( ord_less_rat @ R5 @ ( rep_real2 @ X3 @ N4 ) ) ) ) ) ) ).

% Real.positive.rep_eq
thf(fact_9105_cauchy__inverse,axiom,
    ! [X5: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ~ ( vanishes @ X5 )
       => ( cauchy
          @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X5 @ N4 ) ) ) ) ) ).

% cauchy_inverse
thf(fact_9106_realrel__refl,axiom,
    ! [X5: nat > rat] :
      ( ( cauchy @ X5 )
     => ( realrel @ X5 @ X5 ) ) ).

% realrel_refl
thf(fact_9107_cauchy__add,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( cauchy @ Y6 )
       => ( cauchy
          @ ^ [N4: nat] : ( plus_plus_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ).

% cauchy_add
thf(fact_9108_cauchy__const,axiom,
    ! [X: rat] :
      ( cauchy
      @ ^ [N4: nat] : X ) ).

% cauchy_const
thf(fact_9109_cauchy__minus,axiom,
    ! [X5: nat > rat] :
      ( ( cauchy @ X5 )
     => ( cauchy
        @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X5 @ N4 ) ) ) ) ).

% cauchy_minus
thf(fact_9110_cauchy__mult,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( cauchy @ Y6 )
       => ( cauchy
          @ ^ [N4: nat] : ( times_times_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ).

% cauchy_mult
thf(fact_9111_cauchy__diff,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( cauchy @ Y6 )
       => ( cauchy
          @ ^ [N4: nat] : ( minus_minus_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ).

% cauchy_diff
thf(fact_9112_Real__induct,axiom,
    ! [P: real > $o,X: real] :
      ( ! [X10: nat > rat] :
          ( ( cauchy @ X10 )
         => ( P @ ( real2 @ X10 ) ) )
     => ( P @ X ) ) ).

% Real_induct
thf(fact_9113_cr__real__eq,axiom,
    ( pcr_real
    = ( ^ [X3: nat > rat,Y2: real] :
          ( ( cauchy @ X3 )
          & ( ( real2 @ X3 )
            = Y2 ) ) ) ) ).

% cr_real_eq
thf(fact_9114_cauchy__imp__bounded,axiom,
    ! [X5: nat > rat] :
      ( ( cauchy @ X5 )
     => ? [B5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ B5 )
          & ! [N6: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X5 @ N6 ) ) @ B5 ) ) ) ).

% cauchy_imp_bounded
thf(fact_9115_less__RealD,axiom,
    ! [Y6: nat > rat,X: real] :
      ( ( cauchy @ Y6 )
     => ( ( ord_less_real @ X @ ( real2 @ Y6 ) )
       => ? [N2: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y6 @ N2 ) ) ) ) ) ).

% less_RealD
thf(fact_9116_le__RealI,axiom,
    ! [Y6: nat > rat,X: real] :
      ( ( cauchy @ Y6 )
     => ( ! [N2: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y6 @ N2 ) ) )
       => ( ord_less_eq_real @ X @ ( real2 @ Y6 ) ) ) ) ).

% le_RealI
thf(fact_9117_Real__leI,axiom,
    ! [X5: nat > rat,Y: real] :
      ( ( cauchy @ X5 )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X5 @ N2 ) ) @ Y )
       => ( ord_less_eq_real @ ( real2 @ X5 ) @ Y ) ) ) ).

% Real_leI
thf(fact_9118_minus__Real,axiom,
    ! [X5: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( uminus_uminus_real @ ( real2 @ X5 ) )
        = ( real2
          @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X5 @ N4 ) ) ) ) ) ).

% minus_Real
thf(fact_9119_add__Real,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( cauchy @ Y6 )
       => ( ( plus_plus_real @ ( real2 @ X5 ) @ ( real2 @ Y6 ) )
          = ( real2
            @ ^ [N4: nat] : ( plus_plus_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ) ).

% add_Real
thf(fact_9120_mult__Real,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( cauchy @ Y6 )
       => ( ( times_times_real @ ( real2 @ X5 ) @ ( real2 @ Y6 ) )
          = ( real2
            @ ^ [N4: nat] : ( times_times_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ) ).

% mult_Real
thf(fact_9121_diff__Real,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( cauchy @ Y6 )
       => ( ( minus_minus_real @ ( real2 @ X5 ) @ ( real2 @ Y6 ) )
          = ( real2
            @ ^ [N4: nat] : ( minus_minus_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ) ).

% diff_Real
thf(fact_9122_realrelI,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( cauchy @ Y6 )
       => ( ( vanishes
            @ ^ [N4: nat] : ( minus_minus_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) )
         => ( realrel @ X5 @ Y6 ) ) ) ) ).

% realrelI
thf(fact_9123_eq__Real,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( cauchy @ Y6 )
       => ( ( ( real2 @ X5 )
            = ( real2 @ Y6 ) )
          = ( vanishes
            @ ^ [N4: nat] : ( minus_minus_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ) ).

% eq_Real
thf(fact_9124_vanishes__diff__inverse,axiom,
    ! [X5: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ~ ( vanishes @ X5 )
       => ( ( cauchy @ Y6 )
         => ( ~ ( vanishes @ Y6 )
           => ( ( vanishes
                @ ^ [N4: nat] : ( minus_minus_rat @ ( X5 @ N4 ) @ ( Y6 @ N4 ) ) )
             => ( vanishes
                @ ^ [N4: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X5 @ N4 ) ) @ ( inverse_inverse_rat @ ( Y6 @ N4 ) ) ) ) ) ) ) ) ) ).

% vanishes_diff_inverse
thf(fact_9125_realrel__def,axiom,
    ( realrel
    = ( ^ [X8: nat > rat,Y7: nat > rat] :
          ( ( cauchy @ X8 )
          & ( cauchy @ Y7 )
          & ( vanishes
            @ ^ [N4: nat] : ( minus_minus_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).

% realrel_def
thf(fact_9126_cauchy__not__vanishes__cases,axiom,
    ! [X5: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ~ ( vanishes @ X5 )
       => ? [B5: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B5 )
            & ? [K2: nat] :
                ( ! [N6: nat] :
                    ( ( ord_less_eq_nat @ K2 @ N6 )
                   => ( ord_less_rat @ B5 @ ( uminus_uminus_rat @ ( X5 @ N6 ) ) ) )
                | ! [N6: nat] :
                    ( ( ord_less_eq_nat @ K2 @ N6 )
                   => ( ord_less_rat @ B5 @ ( X5 @ N6 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
thf(fact_9127_positive__Real,axiom,
    ! [X5: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( positive @ ( real2 @ X5 ) )
        = ( ? [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
              & ? [K3: nat] :
                ! [N4: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N4 )
                 => ( ord_less_rat @ R5 @ ( X5 @ N4 ) ) ) ) ) ) ) ).

% positive_Real
thf(fact_9128_cauchy__not__vanishes,axiom,
    ! [X5: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ~ ( vanishes @ X5 )
       => ? [B5: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B5 )
            & ? [K2: nat] :
              ! [N6: nat] :
                ( ( ord_less_eq_nat @ K2 @ N6 )
               => ( ord_less_rat @ B5 @ ( abs_abs_rat @ ( X5 @ N6 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes
thf(fact_9129_cauchy__def,axiom,
    ( cauchy
    = ( ^ [X8: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K3: nat] :
            ! [M3: nat] :
              ( ( ord_less_eq_nat @ K3 @ M3 )
             => ! [N4: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N4 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) ) @ R5 ) ) ) ) ) ) ).

% cauchy_def
thf(fact_9130_cauchyI,axiom,
    ! [X5: nat > rat] :
      ( ! [R4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R4 )
         => ? [K8: nat] :
            ! [M4: nat] :
              ( ( ord_less_eq_nat @ K8 @ M4 )
             => ! [N2: nat] :
                  ( ( ord_less_eq_nat @ K8 @ N2 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) ) @ R4 ) ) ) )
     => ( cauchy @ X5 ) ) ).

% cauchyI
thf(fact_9131_cauchyD,axiom,
    ! [X5: nat > rat,R2: rat] :
      ( ( cauchy @ X5 )
     => ( ( ord_less_rat @ zero_zero_rat @ R2 )
       => ? [K2: nat] :
          ! [M: nat] :
            ( ( ord_less_eq_nat @ K2 @ M )
           => ! [N6: nat] :
                ( ( ord_less_eq_nat @ K2 @ N6 )
               => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X5 @ M ) @ ( X5 @ N6 ) ) ) @ R2 ) ) ) ) ) ).

% cauchyD
thf(fact_9132_inverse__Real,axiom,
    ! [X5: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( ( vanishes @ X5 )
         => ( ( inverse_inverse_real @ ( real2 @ X5 ) )
            = zero_zero_real ) )
        & ( ~ ( vanishes @ X5 )
         => ( ( inverse_inverse_real @ ( real2 @ X5 ) )
            = ( real2
              @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X5 @ N4 ) ) ) ) ) ) ) ).

% inverse_Real
thf(fact_9133_not__positive__Real,axiom,
    ! [X5: nat > rat] :
      ( ( cauchy @ X5 )
     => ( ( ~ ( positive @ ( real2 @ X5 ) ) )
        = ( ! [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
             => ? [K3: nat] :
                ! [N4: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N4 )
                 => ( ord_less_eq_rat @ ( X5 @ N4 ) @ R5 ) ) ) ) ) ) ).

% not_positive_Real
thf(fact_9134_inverse__real__def,axiom,
    ( inverse_inverse_real
    = ( map_fu7146612038024189824t_real @ rep_real2 @ real2
      @ ^ [X8: nat > rat] :
          ( if_nat_rat @ ( vanishes @ X8 )
          @ ^ [N4: nat] : zero_zero_rat
          @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) ) ) ) ).

% inverse_real_def
thf(fact_9135_cr__real__def,axiom,
    ( cr_real
    = ( ^ [X3: nat > rat,Y2: real] :
          ( ( realrel @ X3 @ X3 )
          & ( ( real2 @ X3 )
            = Y2 ) ) ) ) ).

% cr_real_def
thf(fact_9136_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
thf(fact_9137_uminus__real__def,axiom,
    ( uminus_uminus_real
    = ( map_fu7146612038024189824t_real @ rep_real2 @ real2
      @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) ) ) ) ).

% uminus_real_def
thf(fact_9138_times__real__def,axiom,
    ( times_times_real
    = ( map_fu1532550112467129777l_real @ rep_real2 @ ( map_fu7146612038024189824t_real @ rep_real2 @ real2 )
      @ ^ [X8: nat > rat,Y7: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ).

% times_real_def
thf(fact_9139_plus__real__def,axiom,
    ( plus_plus_real
    = ( map_fu1532550112467129777l_real @ rep_real2 @ ( map_fu7146612038024189824t_real @ rep_real2 @ real2 )
      @ ^ [X8: nat > rat,Y7: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ).

% plus_real_def
thf(fact_9140_numeral__le__enat__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( extended_enat2 @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ N ) ) ).

% numeral_le_enat_iff
thf(fact_9141_enat__ord__simps_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% enat_ord_simps(2)
thf(fact_9142_enat__ord__simps_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% enat_ord_simps(1)
thf(fact_9143_idiff__enat__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N )
      = ( extended_enat2 @ zero_zero_nat ) ) ).

% idiff_enat_0
thf(fact_9144_idiff__enat__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
      = N ) ).

% idiff_enat_0_right
thf(fact_9145_numeral__less__enat__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( extended_enat2 @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ N ) ) ).

% numeral_less_enat_iff
thf(fact_9146_one__enat__def,axiom,
    ( one_on7984719198319812577d_enat
    = ( extended_enat2 @ one_one_nat ) ) ).

% one_enat_def
thf(fact_9147_enat__1__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( ( extended_enat2 @ X )
        = one_on7984719198319812577d_enat )
      = ( X = one_one_nat ) ) ).

% enat_1_iff(1)
thf(fact_9148_enat__1__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( one_on7984719198319812577d_enat
        = ( extended_enat2 @ X ) )
      = ( X = one_one_nat ) ) ).

% enat_1_iff(2)
thf(fact_9149_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat
        = ( extended_enat2 @ X ) )
      = ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
thf(fact_9150_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( ( extended_enat2 @ X )
        = zero_z5237406670263579293d_enat )
      = ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
thf(fact_9151_zero__enat__def,axiom,
    ( zero_z5237406670263579293d_enat
    = ( extended_enat2 @ zero_zero_nat ) ) ).

% zero_enat_def
thf(fact_9152_less__enatE,axiom,
    ! [N: extended_enat,M2: nat] :
      ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M2 ) )
     => ~ ! [K2: nat] :
            ( ( N
              = ( extended_enat2 @ K2 ) )
           => ~ ( ord_less_nat @ K2 @ M2 ) ) ) ).

% less_enatE
thf(fact_9153_enat__ile,axiom,
    ! [N: extended_enat,M2: nat] :
      ( ( ord_le2932123472753598470d_enat @ N @ ( extended_enat2 @ M2 ) )
     => ? [K2: nat] :
          ( N
          = ( extended_enat2 @ K2 ) ) ) ).

% enat_ile
thf(fact_9154_finite__enat__bounded,axiom,
    ! [A2: set_Extended_enat,N: nat] :
      ( ! [Y3: extended_enat] :
          ( ( member_Extended_enat @ Y3 @ A2 )
         => ( ord_le2932123472753598470d_enat @ Y3 @ ( extended_enat2 @ N ) ) )
     => ( finite4001608067531595151d_enat @ A2 ) ) ).

% finite_enat_bounded
thf(fact_9155_Suc__ile__eq,axiom,
    ! [M2: nat,N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M2 ) ) @ N )
      = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M2 ) @ N ) ) ).

% Suc_ile_eq
thf(fact_9156_eventually__prod__sequentially,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N3: nat] :
          ! [M3: nat] :
            ( ( ord_less_eq_nat @ N3 @ M3 )
           => ! [N4: nat] :
                ( ( ord_less_eq_nat @ N3 @ N4 )
               => ( P @ ( product_Pair_nat_nat @ N4 @ M3 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_9157_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( extended_enat2 @ N ) )
      = ( ? [Y8: nat,X9: nat] :
            ( ( X
              = ( extended_enat2 @ X9 ) )
            & ( Y
              = ( extended_enat2 @ Y8 ) )
            & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y8 ) @ N ) ) ) ) ).

% iadd_le_enat_iff
thf(fact_9158_elimnum,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        = ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).

% elimnum
thf(fact_9159_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,L: nat] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extended_enat2 @ L ) )
      = ( vEBT_Node @ Info @ Deg
        @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( map_VE8901447254227204932T_VEBT
            @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
            @ TreeList ) )
        @ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide_nat @ L @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.simps(3)
thf(fact_9160_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Uu: extended_enat] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Leaf @ A @ B ) @ Uu )
      = ( vEBT_Leaf @ A @ B ) ) ).

% VEBT_internal.elim_dead.simps(1)
thf(fact_9161_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y
             != ( vEBT_Leaf @ A5 @ B5 ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( Xa2 = extend5688581933313929465d_enat )
               => ( Y
                 != ( vEBT_Node @ Info2 @ Deg2
                    @ ( map_VE8901447254227204932T_VEBT
                      @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      @ TreeList2 )
                    @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ! [L4: nat] :
                    ( ( Xa2
                      = ( extended_enat2 @ L4 ) )
                   => ( Y
                     != ( vEBT_Node @ Info2 @ Deg2
                        @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          @ ( map_VE8901447254227204932T_VEBT
                            @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            @ TreeList2 ) )
                        @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
thf(fact_9162_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ extend5688581933313929465d_enat )
      = ( vEBT_Node @ Info @ Deg
        @ ( map_VE8901447254227204932T_VEBT
          @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          @ TreeList )
        @ ( vEBT_VEBT_elim_dead @ Summary @ extend5688581933313929465d_enat ) ) ) ).

% VEBT_internal.elim_dead.simps(2)
thf(fact_9163_elimcomplete,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ extend5688581933313929465d_enat )
        = ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).

% elimcomplete
thf(fact_9164_enat__ord__code_I3_J,axiom,
    ! [Q4: extended_enat] : ( ord_le2932123472753598470d_enat @ Q4 @ extend5688581933313929465d_enat ) ).

% enat_ord_code(3)
thf(fact_9165_enat__ord__simps_I5_J,axiom,
    ! [Q4: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ Q4 )
      = ( Q4 = extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(5)
thf(fact_9166_times__enat__simps_I4_J,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M2 ) @ extend5688581933313929465d_enat )
          = zero_z5237406670263579293d_enat ) )
      & ( ( M2 != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M2 ) @ extend5688581933313929465d_enat )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
thf(fact_9167_times__enat__simps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
          = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
thf(fact_9168_enat__ord__code_I5_J,axiom,
    ! [N: nat] :
      ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) ) ).

% enat_ord_code(5)
thf(fact_9169_infinity__ileE,axiom,
    ! [M2: nat] :
      ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M2 ) ) ).

% infinity_ileE
thf(fact_9170_enat__add__left__cancel__le,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
      = ( ( A = extend5688581933313929465d_enat )
        | ( ord_le2932123472753598470d_enat @ B @ C ) ) ) ).

% enat_add_left_cancel_le
thf(fact_9171_enat__ord__simps_I3_J,axiom,
    ! [Q4: extended_enat] : ( ord_le2932123472753598470d_enat @ Q4 @ extend5688581933313929465d_enat ) ).

% enat_ord_simps(3)
thf(fact_9172_Sup__enat__def,axiom,
    ( comple4398354569131411667d_enat
    = ( ^ [A6: set_Extended_enat] : ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ zero_z5237406670263579293d_enat @ ( if_Extended_enat @ ( finite4001608067531595151d_enat @ A6 ) @ ( lattic921264341876707157d_enat @ A6 ) @ extend5688581933313929465d_enat ) ) ) ) ).

% Sup_enat_def
thf(fact_9173_Inf__enat__def,axiom,
    ( comple2295165028678016749d_enat
    = ( ^ [A6: set_Extended_enat] :
          ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ extend5688581933313929465d_enat
          @ ( ord_Le1955565732374568822d_enat
            @ ^ [X3: extended_enat] : ( member_Extended_enat @ X3 @ A6 ) ) ) ) ) ).

% Inf_enat_def
thf(fact_9174_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X: produc7272778201969148633d_enat] :
      ( ! [A5: $o,B5: $o,Uu2: extended_enat] :
          ( X
         != ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A5 @ B5 ) @ Uu2 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X
           != ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ extend5688581933313929465d_enat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,L4: nat] :
              ( X
             != ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ).

% VEBT_internal.elim_dead.cases
thf(fact_9175_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X @ Xa2 )
        = Y )
     => ( ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Xa2 = extend5688581933313929465d_enat )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ Deg2
                        @ ( map_VE8901447254227204932T_VEBT
                          @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ TreeList2 )
                        @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) )
                   => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ extend5688581933313929465d_enat ) ) ) ) )
           => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ! [L4: nat] :
                      ( ( Xa2
                        = ( extended_enat2 @ L4 ) )
                     => ( ( Y
                          = ( vEBT_Node @ Info2 @ Deg2
                            @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                              @ ( map_VE8901447254227204932T_VEBT
                                @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                @ TreeList2 ) )
                            @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
thf(fact_9176_times__enat__def,axiom,
    ( times_7803423173614009249d_enat
    = ( ^ [M3: extended_enat,N4: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P5: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P5 ) )
              @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
              @ N4 )
          @ ( if_Extended_enat @ ( N4 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
          @ M3 ) ) ) ).

% times_enat_def
thf(fact_9177_eSuc__Max,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ( extended_eSuc @ ( lattic921264341876707157d_enat @ A2 ) )
          = ( lattic921264341876707157d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A2 ) ) ) ) ) ).

% eSuc_Max
thf(fact_9178_eSuc__ile__mono,axiom,
    ! [N: extended_enat,M2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M2 ) )
      = ( ord_le2932123472753598470d_enat @ N @ M2 ) ) ).

% eSuc_ile_mono
thf(fact_9179_iless__Suc__eq,axiom,
    ! [M2: nat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M2 ) @ ( extended_eSuc @ N ) )
      = ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M2 ) @ N ) ) ).

% iless_Suc_eq
thf(fact_9180_ile__eSuc,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ N @ ( extended_eSuc @ N ) ) ).

% ile_eSuc
thf(fact_9181_not__eSuc__ilei0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_eSuc_ilei0
thf(fact_9182_ileI1,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ M2 @ N )
     => ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ M2 ) @ N ) ) ).

% ileI1
thf(fact_9183_eSuc__Sup,axiom,
    ! [A2: set_Extended_enat] :
      ( ( A2 != bot_bo7653980558646680370d_enat )
     => ( ( extended_eSuc @ ( comple4398354569131411667d_enat @ A2 ) )
        = ( comple4398354569131411667d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A2 ) ) ) ) ).

% eSuc_Sup
thf(fact_9184_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_than_iff
thf(fact_9185_pair__less__def,axiom,
    ( fun_pair_less
    = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).

% pair_less_def
thf(fact_9186_natLeq__on__well__order__on,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( collect_nat
        @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ N )
              & ( ord_less_nat @ Y2 @ N )
              & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).

% natLeq_on_well_order_on
thf(fact_9187_natLeq__on__Well__order,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X3: nat,Y2: nat] :
                ( ( ord_less_nat @ X3 @ N )
                & ( ord_less_nat @ Y2 @ N )
                & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ N )
              & ( ord_less_nat @ Y2 @ N )
              & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).

% natLeq_on_Well_order
thf(fact_9188_Real_Opositive__def,axiom,
    ( positive
    = ( map_fu1856342031159181835at_o_o @ rep_real2 @ id_o
      @ ^ [X8: nat > rat] :
        ? [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
          & ? [K3: nat] :
            ! [N4: nat] :
              ( ( ord_less_eq_nat @ K3 @ N4 )
             => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) ) ) ) ).

% Real.positive_def
thf(fact_9189_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
      = ( ( re @ Z )
        = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% cmod_plus_Re_le_0_iff
thf(fact_9190_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_9191_abs__Re__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% abs_Re_le_cmod
thf(fact_9192_complex__Re__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% complex_Re_le_cmod
thf(fact_9193_complex__abs__le__norm,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% complex_abs_le_norm
thf(fact_9194_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
       => ( ( csqrt @ X )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_9195_abs__Im__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% abs_Im_le_cmod
thf(fact_9196_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( re @ X )
        = ( re @ Y ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).

% cmod_Im_le_iff
thf(fact_9197_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( im @ X )
        = ( im @ Y ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).

% cmod_Re_le_iff
thf(fact_9198_csqrt__principal,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
      | ( ( ( re @ ( csqrt @ Z ) )
          = zero_zero_real )
        & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).

% csqrt_principal
thf(fact_9199_cmod__le,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_le
thf(fact_9200_csqrt__square,axiom,
    ! [B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
        | ( ( ( re @ B )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
     => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = B ) ) ).

% csqrt_square
thf(fact_9201_csqrt__unique,axiom,
    ! [W2: complex,Z: complex] :
      ( ( ( power_power_complex @ W2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = Z )
     => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W2 ) )
          | ( ( ( re @ W2 )
              = zero_zero_real )
            & ( ord_less_eq_real @ zero_zero_real @ ( im @ W2 ) ) ) )
       => ( ( csqrt @ Z )
          = W2 ) ) ) ).

% csqrt_unique
thf(fact_9202_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
       => ( ( csqrt @ X )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_9203_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
        | ( ( ( im @ X )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).

% csqrt_minus
thf(fact_9204_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_9205_Rep__unit__induct,axiom,
    ! [Y: $o,P: $o > $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ! [X4: product_unit] : ( P @ ( product_Rep_unit @ X4 ) )
       => ( P @ Y ) ) ) ).

% Rep_unit_induct
thf(fact_9206_Abs__unit__inject,axiom,
    ! [X: $o,Y: $o] :
      ( ( member_o @ X @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
       => ( ( ( product_Abs_unit @ X )
            = ( product_Abs_unit @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_unit_inject
thf(fact_9207_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
        = Y ) ) ).

% Abs_unit_inverse
thf(fact_9208_Rep__unit,axiom,
    ! [X: product_unit] : ( member_o @ ( product_Rep_unit @ X ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).

% Rep_unit
thf(fact_9209_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y3: $o] :
          ( ( X
            = ( product_Abs_unit @ Y3 ) )
         => ~ ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% Abs_unit_cases
thf(fact_9210_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ~ ! [X4: product_unit] :
            ( Y
            = ( ~ ( product_Rep_unit @ X4 ) ) ) ) ).

% Rep_unit_cases
thf(fact_9211_Abs__unit__induct,axiom,
    ! [P: product_unit > $o,X: product_unit] :
      ( ! [Y3: $o] :
          ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
         => ( P @ ( product_Abs_unit @ Y3 ) ) )
     => ( P @ X ) ) ).

% Abs_unit_induct
thf(fact_9212_type__definition__unit,axiom,
    type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).

% type_definition_unit
thf(fact_9213_Quotient__real,axiom,
    quotie3684837364556693515t_real @ realrel @ real2 @ rep_real2 @ cr_real ).

% Quotient_real
thf(fact_9214_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_9215_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A )
      = A ) ).

% gcd_nat.left_neutral
thf(fact_9216_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_9217_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ zero_zero_nat )
      = A ) ).

% gcd_nat.right_neutral
thf(fact_9218_gcd__0__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ X @ zero_zero_nat )
      = X ) ).

% gcd_0_nat
thf(fact_9219_gcd__0__left__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X )
      = X ) ).

% gcd_0_left_nat
thf(fact_9220_gcd__1__nat,axiom,
    ! [M2: nat] :
      ( ( gcd_gcd_nat @ M2 @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9221_gcd__Suc__0,axiom,
    ! [M2: nat] :
      ( ( gcd_gcd_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9222_gcd__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M2 @ N ) )
      = ( ( M2 != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9223_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X @ Y )
        = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9224_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X3: nat,Y2: nat] : ( if_nat @ ( Y2 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9225_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9226_gcd__diff1__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M2 @ N ) @ N )
        = ( gcd_gcd_nat @ M2 @ N ) ) ) ).

% gcd_diff1_nat
thf(fact_9227_gcd__diff2__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M2 ) @ N )
        = ( gcd_gcd_nat @ M2 @ N ) ) ) ).

% gcd_diff2_nat
thf(fact_9228_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_9229_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_9230_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).

% gcd_ge_0_int
thf(fact_9231_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_9232_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X4: nat,Y3: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X4 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X4 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9233_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X4: nat,Y3: nat] :
          ( ( times_times_nat @ A @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9234_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_9235_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_9236_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( ( ord_less_eq_int @ Y @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9237_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( member_nat @ A5 @ A2 )
         => ( ( member_nat @ B5 @ A2 )
           => ( member_nat @ ( gcd_gcd_nat @ A5 @ B5 ) @ A2 ) ) )
     => ( ( A2 != bot_bot_set_nat )
       => ( member_nat @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).

% Gcd_in
thf(fact_9238_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_9239_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M3: nat,N4: nat] :
        ( ( dvd_dvd_nat @ M3 @ N4 )
        & ( M3 != N4 ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9240_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( gcd_gcd_nat @ M2 @ N )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] :
                ( ( dvd_dvd_nat @ D5 @ M2 )
                & ( dvd_dvd_nat @ D5 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9241_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9242_max__nat_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.semilattice_neutr_axioms
thf(fact_9243_gcd__nat_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.semilattice_neutr_axioms
thf(fact_9244_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
        @ ^ [X3: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_9245_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ) ).

% less_int_def
thf(fact_9246_MOST__nat,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ cofinite_nat )
      = ( ? [M3: nat] :
          ! [N4: nat] :
            ( ( ord_less_nat @ M3 @ N4 )
           => ( P @ N4 ) ) ) ) ).

% MOST_nat
thf(fact_9247_MOST__ge__nat,axiom,
    ! [M2: nat] : ( eventually_nat @ ( ord_less_eq_nat @ M2 ) @ cofinite_nat ) ).

% MOST_ge_nat
thf(fact_9248_MOST__nat__le,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ cofinite_nat )
      = ( ? [M3: nat] :
          ! [N4: nat] :
            ( ( ord_less_eq_nat @ M3 @ N4 )
           => ( P @ N4 ) ) ) ) ).

% MOST_nat_le
thf(fact_9249_MOST__Suc__iff,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [N4: nat] : ( P @ ( suc @ N4 ) )
        @ cofinite_nat )
      = ( eventually_nat @ P @ cofinite_nat ) ) ).

% MOST_Suc_iff
thf(fact_9250_MOST__SucI,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ cofinite_nat )
     => ( eventually_nat
        @ ^ [N4: nat] : ( P @ ( suc @ N4 ) )
        @ cofinite_nat ) ) ).

% MOST_SucI
thf(fact_9251_MOST__SucD,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [N4: nat] : ( P @ ( suc @ N4 ) )
        @ cofinite_nat )
     => ( eventually_nat @ P @ cofinite_nat ) ) ).

% MOST_SucD
thf(fact_9252_gcd__nat_Omonoid__axioms,axiom,
    monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.monoid_axioms
thf(fact_9253_max__nat_Omonoid__axioms,axiom,
    monoid_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.monoid_axioms
thf(fact_9254_INFM__nat,axiom,
    ! [P: nat > $o] :
      ( ( frequently_nat @ P @ cofinite_nat )
      = ( ! [M3: nat] :
          ? [N4: nat] :
            ( ( ord_less_nat @ M3 @ N4 )
            & ( P @ N4 ) ) ) ) ).

% INFM_nat
thf(fact_9255_INFM__nat__le,axiom,
    ! [P: nat > $o] :
      ( ( frequently_nat @ P @ cofinite_nat )
      = ( ! [M3: nat] :
          ? [N4: nat] :
            ( ( ord_less_eq_nat @ M3 @ N4 )
            & ( P @ N4 ) ) ) ) ).

% INFM_nat_le
thf(fact_9256_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs3: list_nat] :
              ( ( X
                = ( cons_nat @ X4 @ Xs3 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_9257_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_9258_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_9259_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_9260_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X4: nat,Xs3: list_nat] :
                ( ( X
                  = ( cons_nat @ X4 @ Xs3 ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs3 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_9261_prod__decode__def,axiom,
    ( nat_prod_decode
    = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).

% prod_decode_def
thf(fact_9262_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 ) )
       => ( ! [N2: nat] :
              ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) )
             => ( ! [X2: nat,Y4: nat] :
                    ( ( ( product_Pair_nat_nat @ X2 @ Y4 )
                      = ( nat_prod_decode @ N2 ) )
                   => ( P @ Y4 ) )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_9263_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N2: nat] :
              ( ( X
                = ( suc @ N2 ) )
             => ( Y
               != ( produc2761476792215241774st_nat
                  @ ^ [X3: nat,Y2: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y2 ) )
                  @ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).

% list_decode.elims
thf(fact_9264_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_9265_list__decode_Osimps_I1_J,axiom,
    ( ( nat_list_decode @ zero_zero_nat )
    = nil_nat ) ).

% list_decode.simps(1)
thf(fact_9266_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( accp_nat @ nat_list_decode_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ N2 ) )
               => ( ( Y
                    = ( produc2761476792215241774st_nat
                      @ ^ [X3: nat,Y2: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y2 ) )
                      @ ( nat_prod_decode @ N2 ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_9267_integer__of__nat__1,axiom,
    ( ( code_integer_of_nat @ one_one_nat )
    = one_one_Code_integer ) ).

% integer_of_nat_1
thf(fact_9268_gcd__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.comm_monoid_axioms
thf(fact_9269_max__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.comm_monoid_axioms
thf(fact_9270_integer__of__nat__0,axiom,
    ( ( code_integer_of_nat @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% integer_of_nat_0
thf(fact_9271_times__num__def,axiom,
    ( times_times_num
    = ( ^ [M3: num,N4: num] : ( num_of_nat @ ( times_times_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N4 ) ) ) ) ) ).

% times_num_def
thf(fact_9272_nat__of__num__neq__0,axiom,
    ! [X: num] :
      ( ( nat_of_num @ X )
     != zero_zero_nat ) ).

% nat_of_num_neq_0
thf(fact_9273_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_9274_nat__of__num__inc,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( inc @ X ) )
      = ( suc @ ( nat_of_num @ X ) ) ) ).

% nat_of_num_inc
thf(fact_9275_num__eq__iff,axiom,
    ( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
    = ( ^ [X3: num,Y2: num] :
          ( ( nat_of_num @ X3 )
          = ( nat_of_num @ Y2 ) ) ) ) ).

% num_eq_iff
thf(fact_9276_nat__of__num__numeral,axiom,
    nat_of_num = numeral_numeral_nat ).

% nat_of_num_numeral
thf(fact_9277_nat__of__num__inverse,axiom,
    ! [X: num] :
      ( ( num_of_nat @ ( nat_of_num @ X ) )
      = X ) ).

% nat_of_num_inverse
thf(fact_9278_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_9279_nat__of__num__add,axiom,
    ! [X: num,Y: num] :
      ( ( nat_of_num @ ( plus_plus_num @ X @ Y ) )
      = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).

% nat_of_num_add
thf(fact_9280_nat__of__num__code_I1_J,axiom,
    ( ( nat_of_num @ one )
    = one_one_nat ) ).

% nat_of_num_code(1)
thf(fact_9281_less__eq__num__def,axiom,
    ( ord_less_eq_num
    = ( ^ [M3: num,N4: num] : ( ord_less_eq_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N4 ) ) ) ) ).

% less_eq_num_def
thf(fact_9282_nat__of__num__mult,axiom,
    ! [X: num,Y: num] :
      ( ( nat_of_num @ ( times_times_num @ X @ Y ) )
      = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).

% nat_of_num_mult
thf(fact_9283_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_9284_nat__of__num__pos,axiom,
    ! [X: num] : ( ord_less_nat @ zero_zero_nat @ ( nat_of_num @ X ) ) ).

% nat_of_num_pos
thf(fact_9285_less__num__def,axiom,
    ( ord_less_num
    = ( ^ [M3: num,N4: num] : ( ord_less_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N4 ) ) ) ) ).

% less_num_def
thf(fact_9286_nat__of__num_Osimps_I1_J,axiom,
    ( ( nat_of_num @ one )
    = ( suc @ zero_zero_nat ) ) ).

% nat_of_num.simps(1)
thf(fact_9287_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_9288_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_9289_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_9290_plus__num__def,axiom,
    ( plus_plus_num
    = ( ^ [M3: num,N4: num] : ( num_of_nat @ ( plus_plus_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N4 ) ) ) ) ) ).

% plus_num_def
thf(fact_9291_pcr__real__def,axiom,
    ( pcr_real
    = ( relcom2856161143838007533t_real
      @ ( bNF_re4702136315717946289at_rat
        @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
        @ ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
      @ cr_real ) ) ).

% pcr_real_def
thf(fact_9292_real_Odomain,axiom,
    ( ( domainp_nat_rat_real @ pcr_real )
    = ( ^ [X3: nat > rat] :
        ? [Y2: nat > rat] :
          ( ( bNF_re4702136315717946289at_rat
            @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
            @ ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 )
            @ X3
            @ Y2 )
          & ( realrel @ Y2 @ Y2 ) ) ) ) ).

% real.domain
thf(fact_9293_Domainp__pcr__real,axiom,
    ( ( domainp_nat_rat_real @ pcr_real )
    = cauchy ) ).

% Domainp_pcr_real
thf(fact_9294_real_Odomain__eq,axiom,
    ( ( domainp_nat_rat_real @ pcr_real )
    = ( ^ [X3: nat > rat] : ( realrel @ X3 @ X3 ) ) ) ).

% real.domain_eq
thf(fact_9295_real_Odomain__par__left__total,axiom,
    ! [P4: ( nat > rat ) > $o] :
      ( ( left_t2768085380646472630at_rat
        @ ( bNF_re4702136315717946289at_rat
          @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
          @ ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) ) )
     => ( ( bNF_re728719798268516973at_o_o
          @ ( bNF_re4702136315717946289at_rat
            @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
            @ ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
          @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
          @ P4
          @ ^ [X3: nat > rat] : ( realrel @ X3 @ X3 ) )
       => ( ( domainp_nat_rat_real @ pcr_real )
          = P4 ) ) ) ).

% real.domain_par_left_total
thf(fact_9296_Rep__real,axiom,
    ! [X: real] :
      ( member_set_nat_rat @ ( rep_real @ X )
      @ ( collect_set_nat_rat
        @ ^ [C5: set_nat_rat] :
          ? [X3: nat > rat] :
            ( ( realrel @ X3 @ X3 )
            & ( C5
              = ( collect_nat_rat @ ( realrel @ X3 ) ) ) ) ) ) ).

% Rep_real
thf(fact_9297_Rep__real__cases,axiom,
    ! [Y: set_nat_rat] :
      ( ( member_set_nat_rat @ Y
        @ ( collect_set_nat_rat
          @ ^ [C5: set_nat_rat] :
            ? [X3: nat > rat] :
              ( ( realrel @ X3 @ X3 )
              & ( C5
                = ( collect_nat_rat @ ( realrel @ X3 ) ) ) ) ) )
     => ~ ! [X4: real] :
            ( Y
           != ( rep_real @ X4 ) ) ) ).

% Rep_real_cases
thf(fact_9298_Rep__real__inject,axiom,
    ! [X: real,Y: real] :
      ( ( ( rep_real @ X )
        = ( rep_real @ Y ) )
      = ( X = Y ) ) ).

% Rep_real_inject
thf(fact_9299_Rep__real__induct,axiom,
    ! [Y: set_nat_rat,P: set_nat_rat > $o] :
      ( ( member_set_nat_rat @ Y
        @ ( collect_set_nat_rat
          @ ^ [C5: set_nat_rat] :
            ? [X3: nat > rat] :
              ( ( realrel @ X3 @ X3 )
              & ( C5
                = ( collect_nat_rat @ ( realrel @ X3 ) ) ) ) ) )
     => ( ! [X4: real] : ( P @ ( rep_real @ X4 ) )
       => ( P @ Y ) ) ) ).

% Rep_real_induct
thf(fact_9300_Abs__real__inverse,axiom,
    ! [Y: set_nat_rat] :
      ( ( member_set_nat_rat @ Y
        @ ( collect_set_nat_rat
          @ ^ [C5: set_nat_rat] :
            ? [X3: nat > rat] :
              ( ( realrel @ X3 @ X3 )
              & ( C5
                = ( collect_nat_rat @ ( realrel @ X3 ) ) ) ) ) )
     => ( ( rep_real @ ( abs_real @ Y ) )
        = Y ) ) ).

% Abs_real_inverse
thf(fact_9301_rep__real__def,axiom,
    ( rep_real2
    = ( quot_r1730120044975580712at_rat @ rep_real ) ) ).

% rep_real_def
thf(fact_9302_Rep__real__inverse,axiom,
    ! [X: real] :
      ( ( abs_real @ ( rep_real @ X ) )
      = X ) ).

% Rep_real_inverse
thf(fact_9303_Abs__real__cases,axiom,
    ! [X: real] :
      ~ ! [Y3: set_nat_rat] :
          ( ( X
            = ( abs_real @ Y3 ) )
         => ~ ( member_set_nat_rat @ Y3
              @ ( collect_set_nat_rat
                @ ^ [C5: set_nat_rat] :
                  ? [X3: nat > rat] :
                    ( ( realrel @ X3 @ X3 )
                    & ( C5
                      = ( collect_nat_rat @ ( realrel @ X3 ) ) ) ) ) ) ) ).

% Abs_real_cases
thf(fact_9304_Abs__real__induct,axiom,
    ! [P: real > $o,X: real] :
      ( ! [Y3: set_nat_rat] :
          ( ( member_set_nat_rat @ Y3
            @ ( collect_set_nat_rat
              @ ^ [C5: set_nat_rat] :
                ? [X3: nat > rat] :
                  ( ( realrel @ X3 @ X3 )
                  & ( C5
                    = ( collect_nat_rat @ ( realrel @ X3 ) ) ) ) ) )
         => ( P @ ( abs_real @ Y3 ) ) )
     => ( P @ X ) ) ).

% Abs_real_induct
thf(fact_9305_Abs__real__inject,axiom,
    ! [X: set_nat_rat,Y: set_nat_rat] :
      ( ( member_set_nat_rat @ X
        @ ( collect_set_nat_rat
          @ ^ [C5: set_nat_rat] :
            ? [X3: nat > rat] :
              ( ( realrel @ X3 @ X3 )
              & ( C5
                = ( collect_nat_rat @ ( realrel @ X3 ) ) ) ) ) )
     => ( ( member_set_nat_rat @ Y
          @ ( collect_set_nat_rat
            @ ^ [C5: set_nat_rat] :
              ? [X3: nat > rat] :
                ( ( realrel @ X3 @ X3 )
                & ( C5
                  = ( collect_nat_rat @ ( realrel @ X3 ) ) ) ) ) )
       => ( ( ( abs_real @ X )
            = ( abs_real @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_real_inject
thf(fact_9306_Real__def,axiom,
    ( real2
    = ( quot_a3129823074075660125t_real @ realrel @ abs_real ) ) ).

% Real_def
thf(fact_9307_type__definition__real,axiom,
    ( type_d8072115097938612567at_rat @ rep_real @ abs_real
    @ ( collect_set_nat_rat
      @ ^ [C5: set_nat_rat] :
        ? [X3: nat > rat] :
          ( ( realrel @ X3 @ X3 )
          & ( C5
            = ( collect_nat_rat @ ( realrel @ X3 ) ) ) ) ) ) ).

% type_definition_real

% Helper facts (38)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X8: real] : ( P @ X8 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X: nat > rat,Y: nat > rat] :
      ( ( if_nat_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X: nat > rat,Y: nat > rat] :
      ( ( if_nat_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X @ Y )
      = 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,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = 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,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = 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,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    vEBT_invar_vebt @ t @ n ).

thf(conj_1,conjecture,
    vEBT_invar_vebt @ ( vEBT_vebt_delete @ t @ x ) @ n ).

%------------------------------------------------------------------------------