TPTP Problem File: ITP234^3.p

%------------------------------------------------------------------------------
% File     : ITP234^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_InsertCorrectness 00806_052000
% 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    : 0067_VEBT_InsertCorrectness_00806_052000 [Des22]

% Status   : Theorem
% Rating   : 0.50 v8.2.0, 0.23 v8.1.0
% Syntax   : Number of formulae    : 9032 (4087 unt; 882 typ;   0 def)
%            Number of atoms       : 24172 (9833 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 92254 (2519   ~; 426   |;1757   &;77540   @)
%                                         (   0 <=>;10012  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :   81 (  80 usr)
%            Number of type conns  : 4235 (4235   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  805 ( 802 usr;  66 con; 0-5 aty)
%            Number of variables   : 22301 (2208   ^;19443   !; 650   ?;22301   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 21:34:37.584
%------------------------------------------------------------------------------
% Could-be-implicit typings (80)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_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__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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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,
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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__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_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__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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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__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
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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,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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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,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Extended____Nat__Oenat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Extended____Nat__Oenat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Option__Ooption_It__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Extended____Nat__Oenat_J,type,
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thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
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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,
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thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
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thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
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thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    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__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__Extended____Nat__Oenat,type,
    extended_enat: $tType ).

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

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

thf(ty_n_t__Real__Oreal,type,
    real: $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 (802)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: real > 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__Real__Oreal,type,
    archim2898591450579166408c_real: real > real ).

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_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).

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

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

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

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

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

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

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

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

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

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_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__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_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,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
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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__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__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__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__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_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__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__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

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_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
    condit2214826472909112428ve_nat: set_nat > $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_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__Int__Oint,type,
    unique5052692396658037445od_int: num > num > product_prod_int_int ).

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__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_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,
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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__Extended____Nat__Oenat,type,
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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__Real__Oreal,type,
    comm_s7457072308508201937r_real: real > nat > real ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
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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__Real__Oreal,type,
    semiri2265585572941072030t_real: nat > real ).

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_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,
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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_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_001t__Complex__Ocomplex,type,
    finite_card_complex: set_complex > 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__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
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thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Extended____Nat__Oenat,type,
    finite4001608067531595151d_enat: set_Extended_enat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

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

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

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

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

thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
    inj_on_real_real: ( real > real ) > set_real > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    map_fu434086159418415080_int_o: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > $o ) > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > int > int > $o ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).

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

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).

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

thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
    strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
    the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).

thf(sy_c_Fun__Def_Ois__measure_001t__Int__Oint,type,
    fun_is_measure_int: ( int > nat ) > $o ).

thf(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set_Pr8693737435421807431at_nat ).

thf(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set_Pr8693737435421807431at_nat ).

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

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

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

thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
    abs_abs_complex: complex > complex ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: int > int ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Extended____Nat__Oenat_M_Eo_J,type,
    minus_2020553357622893040enat_o: ( extended_enat > $o ) > ( extended_enat > $o ) > extended_enat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
    minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
    minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
    minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
    minus_minus_complex: complex > complex > complex ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
    one_one_complex: complex ).

thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
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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__Int__Oint,type,
    sup_sup_int: int > int > int ).

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__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_001t__Complex__Ocomplex_001t__Extended____Nat__Oenat,type,
    lattic7796887085614042845d_enat: ( complex > extended_enat ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    lattic5364784637807008409ex_nat: ( complex > nat ) > 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__Extended____Nat__Oenat,type,
    lattic1996716550891908761d_enat: ( extended_enat > extended_enat ) > set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
    lattic3845382081240766429at_nat: ( extended_enat > nat ) > 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__Extended____Nat__Oenat,type,
    lattic6042659972569420511d_enat: ( int > extended_enat ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Nat__Onat,type,
    lattic8446286672483414039nt_nat: ( int > nat ) > 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__Extended____Nat__Oenat,type,
    lattic8926238025367240251d_enat: ( nat > extended_enat ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
    lattic7446932960582359483at_nat: ( nat > nat ) > 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__Extended____Nat__Oenat,type,
    lattic9066027731366277983d_enat: ( real > extended_enat ) > set_real > real ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Nat__Onat,type,
    lattic5055836439445974935al_nat: ( real > nat ) > 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_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_Ocount__list_001t__Complex__Ocomplex,type,
    count_list_complex: list_complex > complex > nat ).

thf(sy_c_List_Ocount__list_001t__Extended____Nat__Oenat,type,
    count_101369445342291426d_enat: list_Extended_enat > extended_enat > 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__Real__Oreal,type,
    count_list_real: list_real > real > 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__Int__Oint,type,
    distinct_int: list_int > $o ).

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

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

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__Extended____Nat__Oenat,type,
    find_Extended_enat: ( extended_enat > $o ) > list_Extended_enat > option_Extended_enat ).

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__Real__Oreal,type,
    find_real: ( real > $o ) > list_real > option_real ).

thf(sy_c_List_Ofind_001t__Set__Oset_It__Nat__Onat_J,type,
    find_set_nat: ( set_nat > $o ) > list_set_nat > option_set_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_Ogen__length_001t__Int__Oint,type,
    gen_length_int: nat > list_int > nat ).

thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
    gen_length_nat: nat > list_nat > nat ).

thf(sy_c_List_Ogen__length_001t__VEBT____Definitions__OVEBT,type,
    gen_length_VEBT_VEBT: nat > list_VEBT_VEBT > nat ).

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

thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
    linord1735203802627413978nt_int: ( int > int ) > list_int > list_int ).

thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
    linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).

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

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

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

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

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

thf(sy_c_List_Olist_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__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

thf(sy_c_List_Olist_Oset_001t__Num__Onum,type,
    set_num2: list_num > set_num ).

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_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_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

thf(sy_c_List_Olist__update_001t__Extended____Nat__Oenat,type,
    list_u3071683517702156500d_enat: list_Extended_enat > nat > extended_enat > list_Extended_enat ).

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_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_Olistrel1_001t__Int__Oint,type,
    listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
    listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olistrel1_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    listre4828114922151135584at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).

thf(sy_c_List_Olistrel1_001t__VEBT____Definitions__OVEBT,type,
    listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

thf(sy_c_List_Olistrel1p_001t__Int__Oint,type,
    listrel1p_int: ( int > int > $o ) > list_int > list_int > $o ).

thf(sy_c_List_Olistrel1p_001t__Nat__Onat,type,
    listrel1p_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).

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

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

thf(sy_c_List_Onth_001t__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__Real__Oreal,type,
    nth_real: list_real > nat > real ).

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_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

thf(sy_c_List_Oreplicate_001t__Extended____Nat__Oenat,type,
    replic7216382294607269926d_enat: nat > extended_enat > list_Extended_enat ).

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

thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: nat > nat > list_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__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__Real__Oreal,type,
    semiri5074537144036343181t_real: nat > real ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
    semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
    semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
    semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
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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__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__Int__Oint_Mt__Int__Oint_J_J,type,
    size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > nat ).

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__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__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__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: vEBT_VEBT > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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__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__Real__Oreal,type,
    numeral_numeral_real: num > real ).

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

thf(sy_c_Option_Ooption_ONone_001t__Extended____Nat__Oenat,type,
    none_Extended_enat: option_Extended_enat ).

thf(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
    none_int: option_int ).

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_ONone_001t__Real__Oreal,type,
    none_real: option_real ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
    none_set_nat: option_set_nat ).

thf(sy_c_Option_Ooption_ONone_001t__VEBT____Definitions__OVEBT,type,
    none_VEBT_VEBT: option_VEBT_VEBT ).

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_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
    case_option_num_num: num > ( num > num ) > option_num > num ).

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

thf(sy_c_Option_Ooption_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_Order__Relation_OunderS_001t__Nat__Onat,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Extended____Nat__Oenat_M_Eo_J,type,
    bot_bo1954855461789132331enat_o: extended_enat > $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__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    bot_bot_list_nat_o: list_nat > $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_It__Nat__Onat_J_M_Eo_J,type,
    bot_bot_set_nat_o: set_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__VEBT____Definitions__OVEBT_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bot_bo1565574316222977092_nat_o: vEBT_VEBT > nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__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_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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__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__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    bot_bo1642239108664514429BT_nat: set_Pr7556676689462069481BT_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__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_Oless_001_062_It__Extended____Nat__Oenat_M_Eo_J,type,
    ord_le8499522857272258027enat_o: ( extended_enat > $o ) > ( extended_enat > $o ) > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_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_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__Real__Oreal,type,
    ord_less_real: real > real > $o ).

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__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_It__Extended____Nat__Oenat_M_Eo_J,type,
    ord_le100613205991271927enat_o: ( extended_enat > $o ) > ( extended_enat > $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__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__Real__Oreal_M_Eo_J,type,
    ord_less_eq_real_o: ( real > $o ) > ( real > $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_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__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__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__Real__Oreal,type,
    ord_max_real: real > real > real ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_ma4205026669011143323d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).

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

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_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: ( nat > nat ) > $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_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

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__Complex__Ocomplex,type,
    power_power_complex: complex > nat > complex ).

thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
    power_8040749407984259932d_enat: extended_enat > nat > extended_enat ).

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__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__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,
    produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
    produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc5943733680697469783at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > produc6392793444374437607at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    produc3897820843166775703T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > produc9211091688327510695T_VEBT ).

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__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).

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

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

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

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

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

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

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
    produc2676513652042109336d_enat: ( nat > nat > extended_enat ) > product_prod_nat_nat > extended_enat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
    produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).

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

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
    produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).

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

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

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

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

thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Rings_Odvd__class_Odvd_001t__Extended____Nat__Oenat,type,
    dvd_dv3785147216227455552d_enat: extended_enat > extended_enat > $o ).

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

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

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

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

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

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

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Extended____Nat__Oenat,type,
    zero_n1046097342994218471d_enat: $o > extended_enat ).

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

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

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

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

thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
    suminf_int: ( nat > int ) > int ).

thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
    suminf_nat: ( nat > nat ) > nat ).

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

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

thf(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: ( nat > int ) > $o ).

thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
    summable_nat: ( nat > nat ) > $o ).

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

thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
    sums_complex: ( nat > complex ) > complex > $o ).

thf(sy_c_Series_Osums_001t__Int__Oint,type,
    sums_int: ( nat > int ) > int > $o ).

thf(sy_c_Series_Osums_001t__Nat__Onat,type,
    sums_nat: ( nat > nat ) > nat > $o ).

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

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

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

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

thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
    collect_list_complex: ( list_complex > $o ) > set_list_complex ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
    collec8433460942617342167d_enat: ( list_Extended_enat > $o ) > set_li5464603477888414924d_enat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
    collect_list_int: ( list_int > $o ) > set_list_int ).

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

thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).

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

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

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

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

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    collect_set_complex: ( set_complex > $o ) > set_set_complex ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    collec2260605976452661553d_enat: ( set_Extended_enat > $o ) > set_se7270636423289371942d_enat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
    collect_set_int: ( set_int > $o ) > set_set_int ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

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__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_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__Real__Oreal,type,
    insert_real: real > set_real > set_real ).

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_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__Extended____Nat__Oenat,type,
    set_fo2538466533108834004d_enat: ( nat > extended_enat > extended_enat ) > nat > nat > extended_enat > extended_enat ).

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__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).

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

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

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

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

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_OatMost_001t__Extended____Nat__Oenat,type,
    set_or8332593352340944941d_enat: extended_enat > set_Extended_enat ).

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_OatMost_001t__Real__Oreal,type,
    set_ord_atMost_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_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__Extended____Nat__Oenat,type,
    set_or8419480210114673929d_enat: extended_enat > set_Extended_enat ).

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_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
    set_or5984915006950818249n_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or890127255671739683et_nat: set_nat > set_set_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_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_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
    diffs_complex: ( nat > complex ) > nat > complex ).

thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
    diffs_int: ( nat > int ) > nat > int ).

thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
    diffs_real: ( nat > real ) > nat > real ).

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_Osin_001t__Complex__Ocomplex,type,
    sin_complex: complex > complex ).

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

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

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

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

thf(sy_c_Transfer_Obi__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bi_tot896582865486249351at_int: ( product_prod_nat_nat > int > $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_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_Ohigh,type,
    vEBT_VEBT_high: nat > nat > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

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

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

thf(sy_c_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__Int__Oint_J,type,
    member_list_int: list_int > set_list_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__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6693912407220327184at_nat: produc6392793444374437607at_nat > set_Pr1542805901266377927at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $o ).

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__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_real > $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_deg____,type,
    deg: nat ).

thf(sy_v_i____,type,
    i: nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (8120)
thf(fact_0_True,axiom,
    ( i
    = ( vEBT_VEBT_high @ mi @ na ) ) ).

% True
thf(fact_1_insprop,axiom,
    ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) @ ( vEBT_VEBT_high @ mi @ na ) )
    = ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) ).

% insprop
thf(fact_2_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_3_tc,axiom,
    vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) ).

% tc
thf(fact_4__C161_C,axiom,
    ~ ? [X_1: nat] : ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ X_1 ) ).

% "161"
thf(fact_5_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] :
      ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_6_list__update__id,axiom,
    ! [Xs: list_int,I: nat] :
      ( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_7_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_8_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_9_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,X: int] :
      ( ( I != J )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
        = ( nth_int @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_10_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
        = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_11__C162_C,axiom,
    ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ X_1 ) ).

% "162"
thf(fact_12_list__update__overwrite,axiom,
    ! [Xs: list_int,I: nat,X: int,Y: int] :
      ( ( list_update_int @ ( list_update_int @ Xs @ I @ X ) @ I @ Y )
      = ( list_update_int @ Xs @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_13_list__update__overwrite,axiom,
    ! [Xs: list_nat,I: nat,X: nat,Y: nat] :
      ( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I @ Y )
      = ( list_update_nat @ Xs @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_14_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I @ Y )
      = ( list_u1324408373059187874T_VEBT @ Xs @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_15_nsprop,axiom,
    ( ~ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) )
   => ( summary
      = ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ mi @ na ) ) ) ) ).

% nsprop
thf(fact_16__C11_C,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) ) )
     => ( vEBT_invar_vebt @ X2 @ na ) ) ).

% "11"
thf(fact_17_False,axiom,
    ~ ( ord_less_nat @ mi @ xa ) ).

% False
thf(fact_18_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X3: vEBT_VEBT] :
      ( ( I != I2 )
     => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I2 @ X3 )
        = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X3 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_19_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_int,X: int,X3: int] :
      ( ( I != I2 )
     => ( ( list_update_int @ ( list_update_int @ Xs @ I @ X ) @ I2 @ X3 )
        = ( list_update_int @ ( list_update_int @ Xs @ I2 @ X3 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_20_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_nat,X: nat,X3: nat] :
      ( ( I != I2 )
     => ( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I2 @ X3 )
        = ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ X3 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_21_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N: nat,TreeList: list_VEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ N ) ) @ ( vEBT_VEBT_low @ X4 @ N ) ) ) ) ).

% in_children_def
thf(fact_22__C5_Ohyps_C_I7_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "5.hyps"(7)
thf(fact_23__C12_C,axiom,
    vEBT_invar_vebt @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) ) @ ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ mi @ na ) ) @ summary ) @ m ).

% "12"
thf(fact_24_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).

% not_min_Null_member
thf(fact_25_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_26__C1_C,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "1"
thf(fact_27_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_28_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_29__C5_C,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ).

% "5"
thf(fact_30_abcdef,axiom,
    ord_less_nat @ xa @ mi ).

% abcdef
thf(fact_31_member__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_32__C0_C,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( vEBT_invar_vebt @ X2 @ na ) ) ).

% "0"
thf(fact_33__C163_C,axiom,
    ~ ( vEBT_V8194947554948674370ptions @ summary @ i ) ).

% "163"
thf(fact_34_mimaxrel,axiom,
    ( ( xa != mi )
    & ( xa != ma ) ) ).

% mimaxrel
thf(fact_35__C8_C,axiom,
    ( ( suc @ na )
    = m ) ).

% "8"
thf(fact_36_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_37_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_38_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_39_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_40_order__refl,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ X ) ).

% order_refl
thf(fact_41_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_42_order__refl,axiom,
    ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).

% order_refl
thf(fact_43_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_44_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_45_dual__order_Orefl,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ A ) ).

% dual_order.refl
thf(fact_46_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_47_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_48_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_49_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_50_inthall,axiom,
    ! [Xs: list_Extended_enat,P: extended_enat > $o,N2: nat] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ ( set_Extended_enat2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s3941691890525107288d_enat @ Xs ) )
       => ( P @ ( nth_Extended_enat @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_51_inthall,axiom,
    ! [Xs: list_real,P: real > $o,N2: nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_real2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
       => ( P @ ( nth_real @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_52_inthall,axiom,
    ! [Xs: list_set_nat,P: set_nat > $o,N2: nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
       => ( P @ ( nth_set_nat @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_53_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_54_inthall,axiom,
    ! [Xs: list_int,P: int > $o,N2: nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
       => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_55_inthall,axiom,
    ! [Xs: list_nat,P: nat > $o,N2: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
       => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_56_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% deg_not_0
thf(fact_57__C3_C,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "3"
thf(fact_58_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M: nat,N: nat] :
          ( ( ord_less_eq_nat @ M @ N )
          & ( M != N ) ) ) ) ).

% nat_less_le
thf(fact_59_less__imp__le__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_imp_le_nat
thf(fact_60_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N3: nat] :
          ( X
         != ( plus_plus_nat @ N3 @ N3 ) )
     => ~ ! [N3: nat] :
            ( X
           != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).

% even_odd_cases
thf(fact_61_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_62_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_63_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_64_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% nat.inject
thf(fact_65_mem__Collect__eq,axiom,
    ! [A: extended_enat,P: extended_enat > $o] :
      ( ( member_Extended_enat @ A @ ( collec4429806609662206161d_enat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_68_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_69_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_71_Collect__mem__eq,axiom,
    ! [A2: set_Extended_enat] :
      ( ( collec4429806609662206161d_enat
        @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X4: real] : ( member_real @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X4: int] : ( member_int @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_77_Collect__cong,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_real @ P )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_78_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_79_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_80_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_81_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_82_Suc__less__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_less_eq
thf(fact_83_Suc__mono,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_84_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_85_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_86_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_87_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_88_Suc__le__mono,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N2 @ M2 ) ) ).

% Suc_le_mono
thf(fact_89_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_90_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_91_add__Suc__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% add_Suc_right
thf(fact_92_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_93_add__is__0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_94_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_95_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_96_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) )
      = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_list_update
thf(fact_97_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_list_update
thf(fact_98_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_99_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_100_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_101_add__gr__0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_102_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_103_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
     => ( ( list_update_int @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_less__natE,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ~ ! [Q2: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).

% less_natE
thf(fact_112_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_113_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M: nat] :
            ( N2
            = ( suc @ M ) ) ) ) ).

% gr0_conv_Suc
thf(fact_114_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_115_less__add__Suc1,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_116_less__add__Suc2,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).

% less_add_Suc2
thf(fact_117_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M3: nat] :
          ( N2
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_118_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M: nat,N: nat] :
        ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_119_less__imp__Suc__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ? [K3: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_120_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_121_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_int] :
      ( ( size_size_list_int @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_122_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_123_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
       != ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_124_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
       != ( size_size_list_int @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_125_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_126_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M2
              = ( suc @ J2 ) )
            & ( ord_less_nat @ J2 @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_127_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_128_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_129_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_130_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_131_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X )
       != ( size_size_list_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_132_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_133_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M3: nat] :
          ( N2
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_134_add__eq__self__zero,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = M2 )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_135_add__Suc__shift,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
      = ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_136_Zero__not__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_not_Suc
thf(fact_137_Zero__neq__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_neq_Suc
thf(fact_138_Suc__neq__Zero,axiom,
    ! [M2: nat] :
      ( ( suc @ M2 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_139_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_140_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_141_diff__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N2: nat] :
      ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X5: nat,Y3: nat] :
              ( ( P @ X5 @ Y3 )
             => ( P @ ( suc @ X5 ) @ ( suc @ Y3 ) ) )
         => ( P @ M2 @ N2 ) ) ) ) ).

% diff_induct
thf(fact_142_one__is__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M2 @ N2 ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_143_nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( P @ ( suc @ N3 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_induct
thf(fact_144_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_145_add__is__1,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_146_add__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% add_Suc
thf(fact_147_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_148_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_149_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_150_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_151_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_152_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_153_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_154_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N2 @ M2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
        = ( N2 = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_155_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I4: nat] :
            ( ( J
              = ( suc @ I4 ) )
           => ( P @ I4 ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ J )
             => ( ( P @ ( suc @ I4 ) )
               => ( P @ I4 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_156_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
       => ( ! [I4: nat,J3: nat,K3: nat] :
              ( ( ord_less_nat @ I4 @ J3 )
             => ( ( ord_less_nat @ J3 @ K3 )
               => ( ( P @ I4 @ J3 )
                 => ( ( P @ J3 @ K3 )
                   => ( P @ I4 @ K3 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_157_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_158_Suc__less__SucD,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_less_SucD
thf(fact_159_less__antisym,axiom,
    ! [N2: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N2 @ M2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
       => ( M2 = N2 ) ) ) ).

% less_antisym
thf(fact_160_Suc__less__eq2,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
      = ( ? [M4: nat] :
            ( ( M2
              = ( suc @ M4 ) )
            & ( ord_less_nat @ N2 @ M4 ) ) ) ) ).

% Suc_less_eq2
thf(fact_161_All__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N2 )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
           => ( P @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_162_not__less__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_163_less__Suc__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( M2 = N2 ) ) ) ).

% less_Suc_eq
thf(fact_164_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N2 )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_165_less__SucI,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_166_less__SucE,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M2 @ N2 )
       => ( M2 = N2 ) ) ) ).

% less_SucE
thf(fact_167_Suc__lessI,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( ( suc @ M2 )
         != N2 )
       => ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).

% Suc_lessI
thf(fact_168_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I @ J3 )
           => ( K
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_169_Suc__lessD,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_lessD
thf(fact_170_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I @ J3 )
             => ( K
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_171_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_172_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_173_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_174_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_175_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_176_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_177_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_178_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_179_transitive__stepwise__le,axiom,
    ! [M2: nat,N2: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ! [X5: nat] : ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y3: nat,Z: nat] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z )
               => ( R @ X5 @ Z ) ) )
         => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
           => ( R @ M2 @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_180_nat__induct__at__least,axiom,
    ! [M2: nat,N2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( P @ M2 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M2 @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_181_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
             => ( P @ M5 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_182_not__less__eq__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
      = ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).

% not_less_eq_eq
thf(fact_183_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_184_le__Suc__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
      = ( ( ord_less_eq_nat @ M2 @ N2 )
        | ( M2
          = ( suc @ N2 ) ) ) ) ).

% le_Suc_eq
thf(fact_185_Suc__le__D,axiom,
    ! [N2: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
     => ? [M3: nat] :
          ( M6
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_186_le__SucI,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_187_le__SucE,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ~ ( ord_less_eq_nat @ M2 @ N2 )
       => ( M2
          = ( suc @ N2 ) ) ) ) ).

% le_SucE
thf(fact_188_Suc__leD,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% Suc_leD
thf(fact_189_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M: nat,N: nat] :
        ? [K2: nat] :
          ( N
          = ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_190_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_191_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_192_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_193_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_194_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_195_add__leD2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_196_add__leD1,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% add_leD1
thf(fact_197_le__add2,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).

% le_add2
thf(fact_198_le__add1,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).

% le_add1
thf(fact_199_add__leE,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M2 @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_200_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M5: nat] :
                  ( ( ord_less_nat @ M5 @ N3 )
                  & ~ ( P @ M5 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_201_gr__implies__not0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_202_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_203_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_204_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_205_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_206_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_207_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_208_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_209_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_210_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_211_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ! [I5: nat] :
                ( ( ord_less_eq_nat @ I5 @ K3 )
               => ~ ( P @ I5 ) )
            & ( P @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_212_length__pos__if__in__set,axiom,
    ! [X: extended_enat,Xs: list_Extended_enat] :
      ( ( member_Extended_enat @ X @ ( set_Extended_enat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3941691890525107288d_enat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_213_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_214_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_215_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_216_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_217_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_218_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_219_length__induct,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_220_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_221_subset__code_I1_J,axiom,
    ! [Xs: list_Extended_enat,B: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs ) @ B )
      = ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs ) )
           => ( member_Extended_enat @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_222_subset__code_I1_J,axiom,
    ! [Xs: list_real,B: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B )
      = ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
           => ( member_real @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_223_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
           => ( member_set_nat @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_224_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B )
      = ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( member_VEBT_VEBT @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_225_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_226_subset__code_I1_J,axiom,
    ! [Xs: list_int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
           => ( member_int @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_227_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N2: nat,M2: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N2 @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_228_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > extended_enat,N2: nat,M2: nat] :
      ( ! [N3: nat] : ( ord_le72135733267957522d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_le72135733267957522d_enat @ ( F @ N2 ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N2 @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_229_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N2: nat,M2: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N2 @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_230_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N2: nat,M2: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N2 @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_231_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_232_lift__Suc__mono__less,axiom,
    ! [F: nat > extended_enat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_le72135733267957522d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_le72135733267957522d_enat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_233_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_234_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_235_lift__Suc__antimono__le,axiom,
    ! [F: nat > real,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_236_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_237_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_238_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_239_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_240_lift__Suc__mono__le,axiom,
    ! [F: nat > real,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_241_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_242_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_243_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_244_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_245_le__imp__less__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).

% le_imp_less_Suc
thf(fact_246_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).

% less_eq_Suc_le
thf(fact_247_less__Suc__eq__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_Suc_eq_le
thf(fact_248_le__less__Suc__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
        = ( N2 = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_249_Suc__le__lessD,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_le_lessD
thf(fact_250_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ ( suc @ N3 ) )
                 => ( P @ N3 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_251_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ N3 )
                 => ( P @ ( suc @ N3 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_252_Suc__le__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_le_eq
thf(fact_253_Suc__leI,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_leI
thf(fact_254_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M2: nat,K: nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
         => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_255_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K3 )
               => ~ ( P @ I5 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_256_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_VEBT_VEBT,Z2: list_VEBT_VEBT] : Y4 = Z2 )
    = ( ^ [Xs3: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( ( nth_VEBT_VEBT @ Xs3 @ I3 )
                = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_257_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_int,Z2: list_int] : Y4 = Z2 )
    = ( ^ [Xs3: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs3 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs3 ) )
             => ( ( nth_int @ Xs3 @ I3 )
                = ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_258_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_nat,Z2: list_nat] : Y4 = Z2 )
    = ( ^ [Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_259_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: vEBT_VEBT] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs3: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_260_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: int] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs3: list_int] :
            ( ( ( size_size_list_int @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_int @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_261_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: nat] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_262_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( ( nth_VEBT_VEBT @ Xs @ I4 )
              = ( nth_VEBT_VEBT @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_263_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I4 )
              = ( nth_int @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_264_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I4 )
              = ( nth_nat @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_265_set__update__subsetI,axiom,
    ! [Xs: list_Extended_enat,A2: set_Extended_enat,X: extended_enat,I: nat] :
      ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs ) @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ ( list_u3071683517702156500d_enat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_266_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_267_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_268_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_269_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_270_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_271_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_272_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
           => ( P @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
           => ( P @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_273_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( P @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_274_all__nth__imp__all__set,axiom,
    ! [Xs: list_Extended_enat,P: extended_enat > $o,X: extended_enat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s3941691890525107288d_enat @ Xs ) )
         => ( P @ ( nth_Extended_enat @ Xs @ I4 ) ) )
     => ( ( member_Extended_enat @ X @ ( set_Extended_enat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_275_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: real > $o,X: real] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
         => ( P @ ( nth_real @ Xs @ I4 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_276_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
         => ( P @ ( nth_set_nat @ Xs @ I4 ) ) )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_277_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_278_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: int > $o,X: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I4 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_279_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X: nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I4 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_280_in__set__conv__nth,axiom,
    ! [X: extended_enat,Xs: list_Extended_enat] :
      ( ( member_Extended_enat @ X @ ( set_Extended_enat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3941691890525107288d_enat @ Xs ) )
            & ( ( nth_Extended_enat @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_281_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
            & ( ( nth_real @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_282_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
            & ( ( nth_set_nat @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_283_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_284_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_285_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_286_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ! [X5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_287_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_int,P: int > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_288_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_289_nth__mem,axiom,
    ! [N2: nat,Xs: list_Extended_enat] :
      ( ( ord_less_nat @ N2 @ ( size_s3941691890525107288d_enat @ Xs ) )
     => ( member_Extended_enat @ ( nth_Extended_enat @ Xs @ N2 ) @ ( set_Extended_enat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_290_nth__mem,axiom,
    ! [N2: nat,Xs: list_real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
     => ( member_real @ ( nth_real @ Xs @ N2 ) @ ( set_real2 @ Xs ) ) ) ).

% nth_mem
thf(fact_291_nth__mem,axiom,
    ! [N2: nat,Xs: list_set_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ ( nth_set_nat @ Xs @ N2 ) @ ( set_set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_292_nth__mem,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N2 ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% nth_mem
thf(fact_293_nth__mem,axiom,
    ! [N2: nat,Xs: list_int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( member_int @ ( nth_int @ Xs @ N2 ) @ ( set_int2 @ Xs ) ) ) ).

% nth_mem
thf(fact_294_nth__mem,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_295_order__antisym__conv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_296_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_297_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_298_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_299_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_300_linorder__le__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_301_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_302_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_303_ord__le__eq__subst,axiom,
    ! [A: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_304_ord__le__eq__subst,axiom,
    ! [A: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_305_ord__le__eq__subst,axiom,
    ! [A: real,B2: real,F: real > int,C: int] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_306_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_307_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_308_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_309_ord__le__eq__subst,axiom,
    ! [A: int,B2: int,F: int > real,C: real] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_310_ord__le__eq__subst,axiom,
    ! [A: int,B2: int,F: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_311_ord__le__eq__subst,axiom,
    ! [A: int,B2: int,F: int > int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_312_ord__le__eq__subst,axiom,
    ! [A: real,B2: real,F: real > set_nat,C: set_nat] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_313_ord__eq__le__subst,axiom,
    ! [A: real,F: real > real,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_314_ord__eq__le__subst,axiom,
    ! [A: nat,F: real > nat,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_315_ord__eq__le__subst,axiom,
    ! [A: int,F: real > int,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_316_ord__eq__le__subst,axiom,
    ! [A: real,F: nat > real,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_317_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_318_ord__eq__le__subst,axiom,
    ! [A: int,F: nat > int,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_319_ord__eq__le__subst,axiom,
    ! [A: real,F: int > real,B2: int,C: int] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_320_ord__eq__le__subst,axiom,
    ! [A: nat,F: int > nat,B2: int,C: int] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_321_ord__eq__le__subst,axiom,
    ! [A: int,F: int > int,B2: int,C: int] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_322_ord__eq__le__subst,axiom,
    ! [A: set_nat,F: real > set_nat,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_323_linorder__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_linear
thf(fact_324_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_325_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_326_order__eq__refl,axiom,
    ! [X: real,Y: real] :
      ( ( X = Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% order_eq_refl
thf(fact_327_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_328_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_329_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_330_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_331_order__subst2,axiom,
    ! [A: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_332_order__subst2,axiom,
    ! [A: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_333_order__subst2,axiom,
    ! [A: real,B2: real,F: real > int,C: int] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_334_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_335_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_336_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_337_order__subst2,axiom,
    ! [A: int,B2: int,F: int > real,C: real] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_338_order__subst2,axiom,
    ! [A: int,B2: int,F: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_339_order__subst2,axiom,
    ! [A: int,B2: int,F: int > int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_340_order__subst2,axiom,
    ! [A: real,B2: real,F: real > set_nat,C: set_nat] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_341_order__subst1,axiom,
    ! [A: real,F: real > real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_342_order__subst1,axiom,
    ! [A: real,F: nat > real,B2: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_343_order__subst1,axiom,
    ! [A: real,F: int > real,B2: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_344_order__subst1,axiom,
    ! [A: nat,F: real > nat,B2: real,C: real] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_345_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_346_order__subst1,axiom,
    ! [A: nat,F: int > nat,B2: int,C: int] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_347_order__subst1,axiom,
    ! [A: int,F: real > int,B2: real,C: real] :
      ( ( ord_less_eq_int @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_348_order__subst1,axiom,
    ! [A: int,F: nat > int,B2: nat,C: nat] :
      ( ( ord_less_eq_int @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_349_order__subst1,axiom,
    ! [A: int,F: int > int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_350_order__subst1,axiom,
    ! [A: real,F: set_nat > real,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ! [X5: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_351_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: real,Z2: real] : Y4 = Z2 )
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_eq_real @ A3 @ B3 )
          & ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_352_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_353_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_int,Z2: set_int] : Y4 = Z2 )
    = ( ^ [A3: set_int,B3: set_int] :
          ( ( ord_less_eq_set_int @ A3 @ B3 )
          & ( ord_less_eq_set_int @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_354_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_355_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_356_antisym,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_357_antisym,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_358_antisym,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_359_antisym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_360_antisym,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_361_dual__order_Otrans,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ B2 )
       => ( ord_less_eq_real @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_362_dual__order_Otrans,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A )
     => ( ( ord_less_eq_set_nat @ C @ B2 )
       => ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_363_dual__order_Otrans,axiom,
    ! [B2: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A )
     => ( ( ord_less_eq_set_int @ C @ B2 )
       => ( ord_less_eq_set_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_364_dual__order_Otrans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_365_dual__order_Otrans,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_366_dual__order_Oantisym,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_367_dual__order_Oantisym,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A )
     => ( ( ord_less_eq_set_nat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_368_dual__order_Oantisym,axiom,
    ! [B2: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A )
     => ( ( ord_less_eq_set_int @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_369_dual__order_Oantisym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_370_dual__order_Oantisym,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_371_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: real,Z2: real] : Y4 = Z2 )
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_eq_real @ B3 @ A3 )
          & ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_372_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_373_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_int,Z2: set_int] : Y4 = Z2 )
    = ( ^ [A3: set_int,B3: set_int] :
          ( ( ord_less_eq_set_int @ B3 @ A3 )
          & ( ord_less_eq_set_int @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_374_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_375_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_376_linorder__wlog,axiom,
    ! [P: real > real > $o,A: real,B2: real] :
      ( ! [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: real,B4: real] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_377_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: nat,B4: nat] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_378_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B2: int] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: int,B4: int] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_379_order__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z3 )
       => ( ord_less_eq_real @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_380_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z3 )
       => ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_381_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z3 )
       => ( ord_less_eq_set_int @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_382_order__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_eq_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_383_order__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z3 )
       => ( ord_less_eq_int @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_384_order_Otrans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% order.trans
thf(fact_385_order_Otrans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_386_order_Otrans,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% order.trans
thf(fact_387_order_Otrans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_388_order_Otrans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_389_order__antisym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_390_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_391_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_392_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_393_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_394_ord__le__eq__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_395_ord__le__eq__trans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_396_ord__le__eq__trans,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_397_ord__le__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_398_ord__le__eq__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_399_ord__eq__le__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = B2 )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_400_ord__eq__le__trans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( A = B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_401_ord__eq__le__trans,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( A = B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_402_ord__eq__le__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_403_ord__eq__le__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A = B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_404_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: real,Z2: real] : Y4 = Z2 )
    = ( ^ [X4: real,Y5: real] :
          ( ( ord_less_eq_real @ X4 @ Y5 )
          & ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_405_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [X4: set_nat,Y5: set_nat] :
          ( ( ord_less_eq_set_nat @ X4 @ Y5 )
          & ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_406_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_int,Z2: set_int] : Y4 = Z2 )
    = ( ^ [X4: set_int,Y5: set_int] :
          ( ( ord_less_eq_set_int @ X4 @ Y5 )
          & ( ord_less_eq_set_int @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_407_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_408_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [X4: int,Y5: int] :
          ( ( ord_less_eq_int @ X4 @ Y5 )
          & ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_409_le__cases3,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ( ord_less_eq_real @ X @ Y )
       => ~ ( ord_less_eq_real @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_real @ Y @ X )
         => ~ ( ord_less_eq_real @ X @ Z3 ) )
       => ( ( ( ord_less_eq_real @ X @ Z3 )
           => ~ ( ord_less_eq_real @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_real @ Z3 @ Y )
             => ~ ( ord_less_eq_real @ Y @ X ) )
           => ( ( ( ord_less_eq_real @ Y @ Z3 )
               => ~ ( ord_less_eq_real @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_real @ Z3 @ X )
                 => ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_410_le__cases3,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_411_le__cases3,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z3 ) )
       => ( ( ( ord_less_eq_int @ X @ Z3 )
           => ~ ( ord_less_eq_int @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_int @ Z3 @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z3 )
               => ~ ( ord_less_eq_int @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_int @ Z3 @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_412_nle__le,axiom,
    ! [A: real,B2: real] :
      ( ( ~ ( ord_less_eq_real @ A @ B2 ) )
      = ( ( ord_less_eq_real @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_413_nle__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_414_nle__le,axiom,
    ! [A: int,B2: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B2 ) )
      = ( ( ord_less_eq_int @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_415_set__update__memI,axiom,
    ! [N2: nat,Xs: list_Extended_enat,X: extended_enat] :
      ( ( ord_less_nat @ N2 @ ( size_s3941691890525107288d_enat @ Xs ) )
     => ( member_Extended_enat @ X @ ( set_Extended_enat2 @ ( list_u3071683517702156500d_enat @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_416_set__update__memI,axiom,
    ! [N2: nat,Xs: list_real,X: real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_417_set__update__memI,axiom,
    ! [N2: nat,Xs: list_set_nat,X: set_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_418_set__update__memI,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_419_set__update__memI,axiom,
    ! [N2: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_420_set__update__memI,axiom,
    ! [N2: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_421_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_422_order__less__imp__not__less,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_423_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_424_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_425_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_426_order__less__imp__not__eq2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_427_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_428_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_429_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_430_order__less__imp__not__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_431_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_432_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_433_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_434_linorder__less__linear,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
      | ( X = Y )
      | ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_435_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_436_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_437_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_438_order__less__imp__triv,axiom,
    ! [X: extended_enat,Y: extended_enat,P: $o] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_439_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_440_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_441_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_442_order__less__not__sym,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_443_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_444_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_445_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_446_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_447_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_448_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > int,C: int] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_449_order__less__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_450_order__less__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_451_order__less__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_452_order__less__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_453_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_454_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_455_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_456_order__less__subst1,axiom,
    ! [A: nat,F: extended_enat > nat,B2: extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_457_order__less__subst1,axiom,
    ! [A: nat,F: real > nat,B2: real,C: real] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_458_order__less__subst1,axiom,
    ! [A: nat,F: int > nat,B2: int,C: int] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_459_order__less__subst1,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_460_order__less__subst1,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_461_order__less__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B2: real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_462_order__less__subst1,axiom,
    ! [A: extended_enat,F: int > extended_enat,B2: int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_463_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B2: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_464_order__less__subst1,axiom,
    ! [A: real,F: extended_enat > real,B2: extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_465_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_466_order__less__irrefl,axiom,
    ! [X: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ X @ X ) ).

% order_less_irrefl
thf(fact_467_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_468_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_469_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_470_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_471_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_472_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > int,C: int] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_473_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_474_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_475_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_476_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_477_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_478_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_479_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_480_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_481_ord__eq__less__subst,axiom,
    ! [A: real,F: nat > real,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_482_ord__eq__less__subst,axiom,
    ! [A: int,F: nat > int,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_483_ord__eq__less__subst,axiom,
    ! [A: nat,F: extended_enat > nat,B2: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_484_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_485_ord__eq__less__subst,axiom,
    ! [A: real,F: extended_enat > real,B2: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_486_ord__eq__less__subst,axiom,
    ! [A: int,F: extended_enat > int,B2: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_487_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_488_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: real > extended_enat,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_489_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_490_order__less__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ Z3 )
       => ( ord_le72135733267957522d_enat @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_491_order__less__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z3 )
       => ( ord_less_real @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_492_order__less__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z3 )
       => ( ord_less_int @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_493_order__less__asym_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_494_order__less__asym_H,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ~ ( ord_le72135733267957522d_enat @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_495_order__less__asym_H,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ~ ( ord_less_real @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_496_order__less__asym_H,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ~ ( ord_less_int @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_497_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_498_linorder__neq__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
      = ( ( ord_le72135733267957522d_enat @ X @ Y )
        | ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_499_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_500_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_501_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_502_order__less__asym,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% order_less_asym
thf(fact_503_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_504_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_505_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_506_linorder__neqE,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
     => ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
       => ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_507_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_508_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_509_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_510_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_511_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_512_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_513_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_514_order_Ostrict__implies__not__eq,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_515_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_516_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_517_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_518_dual__order_Ostrict__trans,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A )
     => ( ( ord_le72135733267957522d_enat @ C @ B2 )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_519_dual__order_Ostrict__trans,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ B2 )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_520_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_int @ C @ B2 )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_521_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_522_not__less__iff__gr__or__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_523_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_524_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_525_order_Ostrict__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_526_order_Ostrict__trans,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_527_order_Ostrict__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_528_order_Ostrict__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_529_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ A4 )
       => ( ! [A4: nat,B4: nat] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_530_linorder__less__wlog,axiom,
    ! [P: extended_enat > extended_enat > $o,A: extended_enat,B2: extended_enat] :
      ( ! [A4: extended_enat,B4: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: extended_enat] : ( P @ A4 @ A4 )
       => ( ! [A4: extended_enat,B4: extended_enat] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_531_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B2: real] :
      ( ! [A4: real,B4: real] :
          ( ( ord_less_real @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: real] : ( P @ A4 @ A4 )
       => ( ! [A4: real,B4: real] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_532_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B2: int] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_int @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: int] : ( P @ A4 @ A4 )
       => ( ! [A4: int,B4: int] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_533_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X7: nat] : ( P2 @ X7 ) )
    = ( ^ [P3: nat > $o] :
        ? [N: nat] :
          ( ( P3 @ N )
          & ! [M: nat] :
              ( ( ord_less_nat @ M @ N )
             => ~ ( P3 @ M ) ) ) ) ) ).

% exists_least_iff
thf(fact_534_exists__least__iff,axiom,
    ( ( ^ [P2: extended_enat > $o] :
        ? [X7: extended_enat] : ( P2 @ X7 ) )
    = ( ^ [P3: extended_enat > $o] :
        ? [N: extended_enat] :
          ( ( P3 @ N )
          & ! [M: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M @ N )
             => ~ ( P3 @ M ) ) ) ) ) ).

% exists_least_iff
thf(fact_535_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_536_dual__order_Oirrefl,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ A ) ).

% dual_order.irrefl
thf(fact_537_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_538_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_539_dual__order_Oasym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ~ ( ord_less_nat @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_540_dual__order_Oasym,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A )
     => ~ ( ord_le72135733267957522d_enat @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_541_dual__order_Oasym,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ~ ( ord_less_real @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_542_dual__order_Oasym,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ~ ( ord_less_int @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_543_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_544_linorder__cases,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( X != Y )
       => ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_545_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_546_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_547_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_548_antisym__conv3,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ Y @ X )
     => ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_549_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_550_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_551_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X5: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X5 )
             => ( P @ Y6 ) )
         => ( P @ X5 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_552_less__induct,axiom,
    ! [P: extended_enat > $o,A: extended_enat] :
      ( ! [X5: extended_enat] :
          ( ! [Y6: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ Y6 @ X5 )
             => ( P @ Y6 ) )
         => ( P @ X5 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_553_ord__less__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_554_ord__less__eq__trans,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_555_ord__less__eq__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_556_ord__less__eq__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_557_ord__eq__less__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_558_ord__eq__less__trans,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( A = B2 )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_559_ord__eq__less__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_560_ord__eq__less__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A = B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_561_order_Oasym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order.asym
thf(fact_562_order_Oasym,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ~ ( ord_le72135733267957522d_enat @ B2 @ A ) ) ).

% order.asym
thf(fact_563_order_Oasym,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ~ ( ord_less_real @ B2 @ A ) ) ).

% order.asym
thf(fact_564_order_Oasym,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ~ ( ord_less_int @ B2 @ A ) ) ).

% order.asym
thf(fact_565_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_566_less__imp__neq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_567_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_568_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_569_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z: real] :
          ( ( ord_less_real @ X @ Z )
          & ( ord_less_real @ Z @ Y ) ) ) ).

% dense
thf(fact_570_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).

% gt_ex
thf(fact_571_gt__ex,axiom,
    ! [X: real] :
    ? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).

% gt_ex
thf(fact_572_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).

% gt_ex
thf(fact_573_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).

% lt_ex
thf(fact_574_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M5: nat] :
              ( ( ord_less_nat @ M5 @ N3 )
              & ~ ( P @ M5 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_583_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N3 )
             => ( P @ M5 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_584_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_585_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_586_less__not__refl2,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ N2 @ M2 )
     => ( M2 != N2 ) ) ).

% less_not_refl2
thf(fact_587_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_588_nat__neq__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 != N2 )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( ord_less_nat @ N2 @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_589_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ? [X5: nat] :
            ( ( P @ X5 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_590_nat__le__linear,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
      | ( ord_less_eq_nat @ N2 @ M2 ) ) ).

% nat_le_linear
thf(fact_591_le__antisym,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( M2 = N2 ) ) ) ).

% le_antisym
thf(fact_592_eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 = N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% eq_imp_le
thf(fact_593_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_594_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_595_order__le__imp__less__or__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_596_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_597_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_598_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_599_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_600_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_601_linorder__le__less__linear,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
      | ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_602_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_603_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_604_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_605_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_606_order__less__le__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_607_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_608_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: int > extended_enat,C: extended_enat] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_609_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_610_order__less__le__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_611_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_612_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_613_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_614_order__less__le__subst2,axiom,
    ! [A: extended_enat,B2: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_615_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B2: real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_616_order__less__le__subst1,axiom,
    ! [A: real,F: real > real,B2: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_617_order__less__le__subst1,axiom,
    ! [A: nat,F: real > nat,B2: real,C: real] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_618_order__less__le__subst1,axiom,
    ! [A: int,F: real > int,B2: real,C: real] :
      ( ( ord_less_int @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_619_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_620_order__less__le__subst1,axiom,
    ! [A: real,F: nat > real,B2: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_621_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_622_order__less__le__subst1,axiom,
    ! [A: int,F: nat > int,B2: nat,C: nat] :
      ( ( ord_less_int @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_623_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: int > extended_enat,B2: int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_624_order__less__le__subst1,axiom,
    ! [A: real,F: int > real,B2: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_625_order__le__less__subst2,axiom,
    ! [A: real,B2: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_626_order__le__less__subst2,axiom,
    ! [A: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_627_order__le__less__subst2,axiom,
    ! [A: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_628_order__le__less__subst2,axiom,
    ! [A: real,B2: real,F: real > int,C: int] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_629_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_630_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_631_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_632_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_633_order__le__less__subst2,axiom,
    ! [A: int,B2: int,F: int > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_634_order__le__less__subst2,axiom,
    ! [A: int,B2: int,F: int > real,C: real] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_635_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_636_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_637_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B2: real,C: real] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_638_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: int > extended_enat,B2: int,C: int] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_639_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B2: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_640_order__le__less__subst1,axiom,
    ! [A: real,F: extended_enat > real,B2: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_641_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_642_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B2: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_643_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_644_order__le__less__subst1,axiom,
    ! [A: nat,F: extended_enat > nat,B2: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_645_order__less__le__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le2932123472753598470d_enat @ Y @ Z3 )
       => ( ord_le72135733267957522d_enat @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_646_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z3 )
       => ( ord_less_real @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_647_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z3 )
       => ( ord_less_set_nat @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_648_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z3 )
       => ( ord_less_set_int @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_649_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_650_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z3 )
       => ( ord_less_int @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_651_order__le__less__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ Z3 )
       => ( ord_le72135733267957522d_enat @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_652_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z3 )
       => ( ord_less_real @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_653_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ Y @ Z3 )
       => ( ord_less_set_nat @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_654_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_set_int @ Y @ Z3 )
       => ( ord_less_set_int @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_655_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_656_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z3 )
       => ( ord_less_int @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_657_order__neq__le__trans,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( A != B2 )
     => ( ( ord_le2932123472753598470d_enat @ A @ B2 )
       => ( ord_le72135733267957522d_enat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_658_order__neq__le__trans,axiom,
    ! [A: real,B2: real] :
      ( ( A != B2 )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_659_order__neq__le__trans,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A != B2 )
     => ( ( ord_less_eq_set_nat @ A @ B2 )
       => ( ord_less_set_nat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_660_order__neq__le__trans,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A != B2 )
     => ( ( ord_less_eq_set_int @ A @ B2 )
       => ( ord_less_set_int @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_661_order__neq__le__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != B2 )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_662_order__neq__le__trans,axiom,
    ! [A: int,B2: int] :
      ( ( A != B2 )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_663_order__le__neq__trans,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_le72135733267957522d_enat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_664_order__le__neq__trans,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_665_order__le__neq__trans,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_set_nat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_666_order__le__neq__trans,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_set_int @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_667_order__le__neq__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_668_order__le__neq__trans,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_669_order__less__imp__le,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_670_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_671_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_672_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_673_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_674_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_675_linorder__not__less,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
      = ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_676_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_677_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_678_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_679_linorder__not__le,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ X @ Y ) )
      = ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_680_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_681_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_682_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_683_order__less__le,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [X4: extended_enat,Y5: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_684_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X4: real,Y5: real] :
          ( ( ord_less_eq_real @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_685_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X4: set_nat,Y5: set_nat] :
          ( ( ord_less_eq_set_nat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_686_order__less__le,axiom,
    ( ord_less_set_int
    = ( ^ [X4: set_int,Y5: set_int] :
          ( ( ord_less_eq_set_int @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_687_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_688_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y5: int] :
          ( ( ord_less_eq_int @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_689_order__le__less,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [X4: extended_enat,Y5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_690_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X4: real,Y5: real] :
          ( ( ord_less_real @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_691_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X4: set_nat,Y5: set_nat] :
          ( ( ord_less_set_nat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_692_order__le__less,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X4: set_int,Y5: set_int] :
          ( ( ord_less_set_int @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_693_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_nat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_694_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y5: int] :
          ( ( ord_less_int @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_695_dual__order_Ostrict__implies__order,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A )
     => ( ord_le2932123472753598470d_enat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_696_dual__order_Ostrict__implies__order,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ord_less_eq_real @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_697_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A )
     => ( ord_less_eq_set_nat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_698_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_int,A: set_int] :
      ( ( ord_less_set_int @ B2 @ A )
     => ( ord_less_eq_set_int @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_699_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ord_less_eq_nat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_700_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ord_less_eq_int @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_701_order_Ostrict__implies__order,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ord_le2932123472753598470d_enat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_702_order_Ostrict__implies__order,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_703_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A @ B2 )
     => ( ord_less_eq_set_nat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_704_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A @ B2 )
     => ( ord_less_eq_set_int @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_705_order_Ostrict__implies__order,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_706_order_Ostrict__implies__order,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_707_dual__order_Ostrict__iff__not,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
          & ~ ( ord_le2932123472753598470d_enat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_708_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_less_eq_real @ B3 @ A3 )
          & ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_709_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_710_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [B3: set_int,A3: set_int] :
          ( ( ord_less_eq_set_int @ B3 @ A3 )
          & ~ ( ord_less_eq_set_int @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_711_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_712_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_713_dual__order_Ostrict__trans2,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A )
     => ( ( ord_le2932123472753598470d_enat @ C @ B2 )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_714_dual__order_Ostrict__trans2,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ B2 )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_715_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A )
     => ( ( ord_less_eq_set_nat @ C @ B2 )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_716_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_int,A: set_int,C: set_int] :
      ( ( ord_less_set_int @ B2 @ A )
     => ( ( ord_less_eq_set_int @ C @ B2 )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_717_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_718_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_719_dual__order_Ostrict__trans1,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A )
     => ( ( ord_le72135733267957522d_enat @ C @ B2 )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_720_dual__order_Ostrict__trans1,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_real @ C @ B2 )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_721_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A )
     => ( ( ord_less_set_nat @ C @ B2 )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_722_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A )
     => ( ( ord_less_set_int @ C @ B2 )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_723_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_724_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_int @ C @ B2 )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_725_dual__order_Ostrict__iff__order,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_726_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_less_eq_real @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_727_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_728_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [B3: set_int,A3: set_int] :
          ( ( ord_less_eq_set_int @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_729_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_730_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_731_dual__order_Oorder__iff__strict,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_732_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_less_real @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_733_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_set_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_734_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B3: set_int,A3: set_int] :
          ( ( ord_less_set_int @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_735_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_736_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_int @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_737_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ! [W: real] :
            ( ( ord_less_real @ X @ W )
           => ( ( ord_less_real @ W @ Y )
             => ( ord_less_eq_real @ W @ Z3 ) ) )
       => ( ord_less_eq_real @ Y @ Z3 ) ) ) ).

% dense_le_bounded
thf(fact_738_dense__ge__bounded,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_real @ Z3 @ X )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z3 @ W )
           => ( ( ord_less_real @ W @ X )
             => ( ord_less_eq_real @ Y @ W ) ) )
       => ( ord_less_eq_real @ Y @ Z3 ) ) ) ).

% dense_ge_bounded
thf(fact_739_order_Ostrict__iff__not,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B3 )
          & ~ ( ord_le2932123472753598470d_enat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_740_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_eq_real @ A3 @ B3 )
          & ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_741_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_742_order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [A3: set_int,B3: set_int] :
          ( ( ord_less_eq_set_int @ A3 @ B3 )
          & ~ ( ord_less_eq_set_int @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_743_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_744_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_745_order_Ostrict__trans2,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_746_order_Ostrict__trans2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_747_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_748_order_Ostrict__trans2,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( ord_less_set_int @ A @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_749_order_Ostrict__trans2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_750_order_Ostrict__trans2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_751_order_Ostrict__trans1,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_752_order_Ostrict__trans1,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_753_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_set_nat @ B2 @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_754_order_Ostrict__trans1,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( ord_less_set_int @ B2 @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_755_order_Ostrict__trans1,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_756_order_Ostrict__trans1,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_757_order_Ostrict__iff__order,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_758_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_eq_real @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_759_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_760_order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [A3: set_int,B3: set_int] :
          ( ( ord_less_eq_set_int @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_761_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_762_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_763_order_Oorder__iff__strict,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_764_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_real @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_765_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_set_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_766_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A3: set_int,B3: set_int] :
          ( ( ord_less_set_int @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_767_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_768_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_769_not__le__imp__less,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ~ ( ord_le2932123472753598470d_enat @ Y @ X )
     => ( ord_le72135733267957522d_enat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_770_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_771_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_772_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_773_less__le__not__le,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [X4: extended_enat,Y5: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X4 @ Y5 )
          & ~ ( ord_le2932123472753598470d_enat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_774_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X4: real,Y5: real] :
          ( ( ord_less_eq_real @ X4 @ Y5 )
          & ~ ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_775_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X4: set_nat,Y5: set_nat] :
          ( ( ord_less_eq_set_nat @ X4 @ Y5 )
          & ~ ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_776_less__le__not__le,axiom,
    ( ord_less_set_int
    = ( ^ [X4: set_int,Y5: set_int] :
          ( ( ord_less_eq_set_int @ X4 @ Y5 )
          & ~ ( ord_less_eq_set_int @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_777_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_778_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y5: int] :
          ( ( ord_less_eq_int @ X4 @ Y5 )
          & ~ ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_779_dense__le,axiom,
    ! [Y: real,Z3: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ X5 @ Y )
         => ( ord_less_eq_real @ X5 @ Z3 ) )
     => ( ord_less_eq_real @ Y @ Z3 ) ) ).

% dense_le
thf(fact_780_dense__ge,axiom,
    ! [Z3: real,Y: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ Z3 @ X5 )
         => ( ord_less_eq_real @ Y @ X5 ) )
     => ( ord_less_eq_real @ Y @ Z3 ) ) ).

% dense_ge
thf(fact_781_antisym__conv2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_782_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_783_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_784_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_785_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_786_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_787_antisym__conv1,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le2932123472753598470d_enat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_788_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_789_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_790_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_791_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_792_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_793_nless__le,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ A @ B2 ) )
      = ( ~ ( ord_le2932123472753598470d_enat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_794_nless__le,axiom,
    ! [A: real,B2: real] :
      ( ( ~ ( ord_less_real @ A @ B2 ) )
      = ( ~ ( ord_less_eq_real @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_795_nless__le,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_796_nless__le,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ~ ( ord_less_set_int @ A @ B2 ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_797_nless__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_798_nless__le,axiom,
    ! [A: int,B2: int] :
      ( ( ~ ( ord_less_int @ A @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_799_leI,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).

% leI
thf(fact_800_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_801_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_802_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_803_leD,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X )
     => ~ ( ord_le72135733267957522d_enat @ X @ Y ) ) ).

% leD
thf(fact_804_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_805_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ~ ( ord_less_set_nat @ X @ Y ) ) ).

% leD
thf(fact_806_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ~ ( ord_less_set_int @ X @ Y ) ) ).

% leD
thf(fact_807_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_808_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_809_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_810_le__neq__implies__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( M2 != N2 )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_811_less__or__eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ord_less_nat @ M2 @ N2 )
        | ( M2 = N2 ) )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_812_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M: nat,N: nat] :
          ( ( ord_less_nat @ M @ N )
          | ( M = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_813_buildup__gives__valid,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).

% buildup_gives_valid
thf(fact_814_add__less__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_815_add__less__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_816_add__less__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_817_add__less__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_818_add__less__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_819_add__less__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_820_less__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_821_less__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( ord_less_real @ zero_zero_real @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_822_less__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_823_less__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_824_less__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B2 @ A ) )
      = ( ord_less_real @ zero_zero_real @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_825_less__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_826_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_827_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_828_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_829_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_830_add__le__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_831_add__le__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_832_add__le__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_833_add__le__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_834_add__le__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_835_add__le__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_836_le__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_837_le__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_838_le__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_839_le__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B2 @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_840_le__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_841_le__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B2 @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_842_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_843_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_844_add__left__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_845_add__left__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_846_add__left__cancel,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_847_add__right__cancel,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_848_add__right__cancel,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_849_add__right__cancel,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_850_le__zero__eq,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% le_zero_eq
thf(fact_851_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_852_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_853_not__gr__zero,axiom,
    ! [N2: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% not_gr_zero
thf(fact_854_add__le__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
      = ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_855_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_856_add__le__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
      = ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_857_add__le__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_858_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_859_add__le__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_860_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_861_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_862_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_863_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_864_add_Oright__neutral,axiom,
    ! [A: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ A @ zero_z5237406670263579293d_enat )
      = A ) ).

% add.right_neutral
thf(fact_865_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_866_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_867_add__cancel__left__left,axiom,
    ! [B2: nat,A: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = A )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_868_add__cancel__left__left,axiom,
    ! [B2: real,A: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = A )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_869_add__cancel__left__left,axiom,
    ! [B2: int,A: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = A )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_870_add__cancel__left__left,axiom,
    ! [B2: complex,A: complex] :
      ( ( ( plus_plus_complex @ B2 @ A )
        = A )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_left
thf(fact_871_add__cancel__left__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = A )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_872_add__cancel__left__right,axiom,
    ! [A: real,B2: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = A )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_873_add__cancel__left__right,axiom,
    ! [A: int,B2: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = A )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_874_add__cancel__left__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A @ B2 )
        = A )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_right
thf(fact_875_add__cancel__right__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ B2 @ A ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_876_add__cancel__right__left,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( plus_plus_real @ B2 @ A ) )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_877_add__cancel__right__left,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ B2 @ A ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_878_add__cancel__right__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A
        = ( plus_plus_complex @ B2 @ A ) )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_left
thf(fact_879_add__cancel__right__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_880_add__cancel__right__right,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( plus_plus_real @ A @ B2 ) )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_881_add__cancel__right__right,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ A @ B2 ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_882_add__cancel__right__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( A
        = ( plus_plus_complex @ A @ B2 ) )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_right
thf(fact_883_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_884_add__eq__0__iff__both__eq__0,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
        = zero_z5237406670263579293d_enat )
      = ( ( X = zero_z5237406670263579293d_enat )
        & ( Y = zero_z5237406670263579293d_enat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_885_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_886_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( zero_z5237406670263579293d_enat
        = ( plus_p3455044024723400733d_enat @ X @ Y ) )
      = ( ( X = zero_z5237406670263579293d_enat )
        & ( Y = zero_z5237406670263579293d_enat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_887_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_888_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_889_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_890_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_891_add__0,axiom,
    ! [A: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ zero_z5237406670263579293d_enat @ A )
      = A ) ).

% add_0
thf(fact_892_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_nat @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_893_add__less__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_894_add__less__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_895_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_nat @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_896_add__less__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_897_add__less__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_898_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_899_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_900_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_901_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_902_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_903_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
      = ( X = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_904_zero__reorient,axiom,
    ! [X: extended_enat] :
      ( ( zero_z5237406670263579293d_enat = X )
      = ( X = zero_z5237406670263579293d_enat ) ) ).

% zero_reorient
thf(fact_905_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_906_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_907_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_908_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ C )
      = ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_909_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_910_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_911_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_912_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_p3455044024723400733d_enat @ I @ K )
        = ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_913_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B2: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_914_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B2: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_915_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B2: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A2 @ B2 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_916_group__cancel_Oadd1,axiom,
    ! [A2: extended_enat,K: extended_enat,A: extended_enat,B2: extended_enat] :
      ( ( A2
        = ( plus_p3455044024723400733d_enat @ K @ A ) )
     => ( ( plus_p3455044024723400733d_enat @ A2 @ B2 )
        = ( plus_p3455044024723400733d_enat @ K @ ( plus_p3455044024723400733d_enat @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_917_group__cancel_Oadd2,axiom,
    ! [B: nat,K: nat,B2: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ K @ B2 ) )
     => ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_918_group__cancel_Oadd2,axiom,
    ! [B: int,K: int,B2: int,A: int] :
      ( ( B
        = ( plus_plus_int @ K @ B2 ) )
     => ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_919_group__cancel_Oadd2,axiom,
    ! [B: real,K: real,B2: real,A: real] :
      ( ( B
        = ( plus_plus_real @ K @ B2 ) )
     => ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_920_group__cancel_Oadd2,axiom,
    ! [B: extended_enat,K: extended_enat,B2: extended_enat,A: extended_enat] :
      ( ( B
        = ( plus_p3455044024723400733d_enat @ K @ B2 ) )
     => ( ( plus_p3455044024723400733d_enat @ A @ B )
        = ( plus_p3455044024723400733d_enat @ K @ ( plus_p3455044024723400733d_enat @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_921_add_Oassoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_922_add_Oassoc,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_923_add_Oassoc,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_924_add_Oassoc,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ C )
      = ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_925_add_Oleft__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ A @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_926_add_Oleft__cancel,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ A @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_927_add_Oright__cancel,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_928_add_Oright__cancel,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_929_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_930_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_931_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_932_add_Ocommute,axiom,
    ( plus_p3455044024723400733d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] : ( plus_p3455044024723400733d_enat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_933_add_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_934_add_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_935_add_Oleft__commute,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( plus_plus_real @ B2 @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_936_add_Oleft__commute,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ B2 @ ( plus_p3455044024723400733d_enat @ A @ C ) )
      = ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_937_add__left__imp__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_938_add__left__imp__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_939_add__left__imp__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_940_add__right__imp__eq,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_941_add__right__imp__eq,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_942_add__right__imp__eq,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_943_zero__le,axiom,
    ! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X ) ).

% zero_le
thf(fact_944_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_945_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_946_gr__zeroI,axiom,
    ! [N2: extended_enat] :
      ( ( N2 != zero_z5237406670263579293d_enat )
     => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ).

% gr_zeroI
thf(fact_947_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_948_not__less__zero,axiom,
    ! [N2: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).

% not_less_zero
thf(fact_949_gr__implies__not__zero,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_950_gr__implies__not__zero,axiom,
    ! [M2: extended_enat,N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ M2 @ N2 )
     => ( N2 != zero_z5237406670263579293d_enat ) ) ).

% gr_implies_not_zero
thf(fact_951_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_952_zero__less__iff__neq__zero,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = ( N2 != zero_z5237406670263579293d_enat ) ) ).

% zero_less_iff_neq_zero
thf(fact_953_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( ord_le2932123472753598470d_enat @ I @ J )
        & ( K = L ) )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_954_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_955_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_956_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_957_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( I = J )
        & ( ord_le2932123472753598470d_enat @ K @ L ) )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_958_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_959_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_960_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_961_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( ord_le2932123472753598470d_enat @ I @ J )
        & ( ord_le2932123472753598470d_enat @ K @ L ) )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_962_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_963_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_964_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_965_add__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ C @ D )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_966_add__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_967_add__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_968_add__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_969_add__left__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ C @ A ) @ ( plus_p3455044024723400733d_enat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_970_add__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_971_add__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_972_add__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_973_less__eqE,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ~ ! [C2: extended_enat] :
            ( B2
           != ( plus_p3455044024723400733d_enat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_974_less__eqE,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ~ ! [C2: nat] :
            ( B2
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_975_add__right__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_976_add__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_977_add__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_978_add__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_979_le__iff__add,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
        ? [C3: extended_enat] :
          ( B3
          = ( plus_p3455044024723400733d_enat @ A3 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_980_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
        ? [C3: nat] :
          ( B3
          = ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_981_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_982_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_983_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_984_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_985_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_986_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_987_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_988_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_989_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_990_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_991_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ zero_z5237406670263579293d_enat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_992_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_993_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_994_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_995_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_996_add_Ocomm__neutral,axiom,
    ! [A: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ A @ zero_z5237406670263579293d_enat )
      = A ) ).

% add.comm_neutral
thf(fact_997_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_998_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_999_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1000_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_1001_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_1002_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_1003_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_1004_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_1005_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_1006_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_1007_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_1008_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_1009_add__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1010_add__strict__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_le72135733267957522d_enat @ C @ D )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1011_add__strict__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1012_add__strict__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1013_add__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_1014_add__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_1015_add__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_1016_add__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1017_add__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1018_add__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1019_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_nat @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_1020_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
     => ( ord_less_real @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_1021_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
     => ( ord_less_int @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_1022_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_nat @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_1023_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
     => ( ord_less_real @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_1024_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
     => ( ord_less_int @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_1025_add__nonpos__eq__0__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ Y @ zero_z5237406670263579293d_enat )
       => ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
            = zero_z5237406670263579293d_enat )
          = ( ( X = zero_z5237406670263579293d_enat )
            & ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1026_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_1027_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_1028_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_1029_add__nonneg__eq__0__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ Y )
       => ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
            = zero_z5237406670263579293d_enat )
          = ( ( X = zero_z5237406670263579293d_enat )
            & ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1030_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_1031_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_1032_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_1033_add__nonpos__nonpos,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ zero_z5237406670263579293d_enat )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1034_add__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_1035_add__nonpos__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1036_add__nonpos__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_1037_add__nonneg__nonneg,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B2 )
       => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1038_add__nonneg__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1039_add__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1040_add__nonneg__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1041_add__increasing2,axiom,
    ! [C: extended_enat,B2: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ A )
       => ( ord_le2932123472753598470d_enat @ B2 @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1042_add__increasing2,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B2 @ A )
       => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1043_add__increasing2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1044_add__increasing2,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B2 @ A )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1045_add__decreasing2,axiom,
    ! [C: extended_enat,A: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ A @ B2 )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_1046_add__decreasing2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_1047_add__decreasing2,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_1048_add__decreasing2,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_1049_add__increasing,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ord_le2932123472753598470d_enat @ B2 @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1050_add__increasing,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1051_add__increasing,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1052_add__increasing,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1053_add__decreasing,axiom,
    ! [A: extended_enat,C: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ C @ B2 )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_1054_add__decreasing,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_1055_add__decreasing,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_1056_add__decreasing,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_add__le__less__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1064_add__le__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1065_add__le__less__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1066_add__less__le__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1067_add__less__le__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1068_add__less__le__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1069_pos__add__strict,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1070_pos__add__strict,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ord_le72135733267957522d_enat @ B2 @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1071_pos__add__strict,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1072_pos__add__strict,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1073_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ! [C2: nat] :
            ( ( B2
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1074_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ~ ! [C2: extended_enat] :
            ( ( B2
              = ( plus_p3455044024723400733d_enat @ A @ C2 ) )
           => ( C2 = zero_z5237406670263579293d_enat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1075_add__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_1076_add__pos__pos,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ B2 )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_1077_add__pos__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_1078_add__pos__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_1079_add__neg__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_1080_add__neg__neg,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le72135733267957522d_enat @ B2 @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_neg_neg
thf(fact_1081_add__neg__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_1082_add__neg__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_1083_add__neg__nonpos,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_neg_nonpos
thf(fact_1084_add__neg__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_1085_add__neg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_1086_add__neg__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_1087_add__nonneg__pos,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ B2 )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_1088_add__nonneg__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_1089_add__nonneg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_1090_add__nonneg__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_1091_add__nonpos__neg,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le72135733267957522d_enat @ B2 @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_nonpos_neg
thf(fact_1092_add__nonpos__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_1093_add__nonpos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_1094_add__nonpos__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_1095_add__pos__nonneg,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B2 )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_1096_add__pos__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_1097_add__pos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_1098_add__pos__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_1099_add__strict__increasing,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1100_add__strict__increasing,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1101_add__strict__increasing,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1102_add__strict__increasing2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1103_add__strict__increasing2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1104_add__strict__increasing2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1105_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_1106_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_1107_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_1108_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_1109_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_1110_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_1111_buildup__gives__empty,axiom,
    ! [N2: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_1112_subsetI,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( member_Extended_enat @ X5 @ B ) )
     => ( ord_le7203529160286727270d_enat @ A2 @ B ) ) ).

% subsetI
thf(fact_1113_subsetI,axiom,
    ! [A2: set_real,B: set_real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( member_real @ X5 @ B ) )
     => ( ord_less_eq_set_real @ A2 @ B ) ) ).

% subsetI
thf(fact_1114_subsetI,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ A2 )
         => ( member_set_nat @ X5 @ B ) )
     => ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ).

% subsetI
thf(fact_1115_subsetI,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( member_nat @ X5 @ B ) )
     => ( ord_less_eq_set_nat @ A2 @ B ) ) ).

% subsetI
thf(fact_1116_subsetI,axiom,
    ! [A2: set_int,B: set_int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( member_int @ X5 @ B ) )
     => ( ord_less_eq_set_int @ A2 @ B ) ) ).

% subsetI
thf(fact_1117_psubsetI,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( A2 != B )
       => ( ord_less_set_nat @ A2 @ B ) ) ) ).

% psubsetI
thf(fact_1118_psubsetI,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B )
     => ( ( A2 != B )
       => ( ord_less_set_int @ A2 @ B ) ) ) ).

% psubsetI
thf(fact_1119_subset__antisym,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( ord_less_eq_set_nat @ B @ A2 )
       => ( A2 = B ) ) ) ).

% subset_antisym
thf(fact_1120_subset__antisym,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B )
     => ( ( ord_less_eq_set_int @ B @ A2 )
       => ( A2 = B ) ) ) ).

% subset_antisym
thf(fact_1121_buildup__nothing__in__min__max,axiom,
    ! [N2: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_1122_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_1123_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X4: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X4 )
          | ( vEBT_VEBT_membermima @ T2 @ X4 ) ) ) ) ).

% both_member_options_def
thf(fact_1124_empty__Collect__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( bot_bot_set_list_nat
        = ( collect_list_nat @ P ) )
      = ( ! [X4: list_nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1125_empty__Collect__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P ) )
      = ( ! [X4: set_nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1126_empty__Collect__eq,axiom,
    ! [P: extended_enat > $o] :
      ( ( bot_bo7653980558646680370d_enat
        = ( collec4429806609662206161d_enat @ P ) )
      = ( ! [X4: extended_enat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1127_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X4: real] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1128_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X4: nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1129_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X4: int] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1130_Collect__empty__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( ! [X4: list_nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1131_Collect__empty__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( ! [X4: set_nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1132_Collect__empty__eq,axiom,
    ! [P: extended_enat > $o] :
      ( ( ( collec4429806609662206161d_enat @ P )
        = bot_bo7653980558646680370d_enat )
      = ( ! [X4: extended_enat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1133_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X4: real] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1134_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1135_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1136_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( ! [X4: set_nat] :
            ~ ( member_set_nat @ X4 @ A2 ) )
      = ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_1137_all__not__in__conv,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ! [X4: extended_enat] :
            ~ ( member_Extended_enat @ X4 @ A2 ) )
      = ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% all_not_in_conv
thf(fact_1138_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ( ! [X4: real] :
            ~ ( member_real @ X4 @ A2 ) )
      = ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_1139_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ! [X4: nat] :
            ~ ( member_nat @ X4 @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_1140_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ! [X4: int] :
            ~ ( member_int @ X4 @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_1141_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_1142_empty__iff,axiom,
    ! [C: extended_enat] :
      ~ ( member_Extended_enat @ C @ bot_bo7653980558646680370d_enat ) ).

% empty_iff
thf(fact_1143_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_1144_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_1145_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_1146_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N2 )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_1147_empty__subsetI,axiom,
    ! [A2: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ bot_bo7653980558646680370d_enat @ A2 ) ).

% empty_subsetI
thf(fact_1148_empty__subsetI,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% empty_subsetI
thf(fact_1149_empty__subsetI,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% empty_subsetI
thf(fact_1150_empty__subsetI,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% empty_subsetI
thf(fact_1151_subset__empty,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ bot_bo7653980558646680370d_enat )
      = ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% subset_empty
thf(fact_1152_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_1153_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_1154_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_1155_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% List.finite_set
thf(fact_1156_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_1157_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).

% List.finite_set
thf(fact_1158_List_Ofinite__set,axiom,
    ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).

% List.finite_set
thf(fact_1159_List_Ofinite__set,axiom,
    ! [Xs: list_Extended_enat] : ( finite4001608067531595151d_enat @ ( set_Extended_enat2 @ Xs ) ) ).

% List.finite_set
thf(fact_1160_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ S2 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S2 )
                  & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1161_ex__min__if__finite,axiom,
    ! [S2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ S2 )
            & ~ ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ S2 )
                  & ( ord_le72135733267957522d_enat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1162_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ S2 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S2 )
                  & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1163_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ S2 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S2 )
                  & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1164_not__psubset__empty,axiom,
    ! [A2: set_Extended_enat] :
      ~ ( ord_le2529575680413868914d_enat @ A2 @ bot_bo7653980558646680370d_enat ) ).

% not_psubset_empty
thf(fact_1165_not__psubset__empty,axiom,
    ! [A2: set_real] :
      ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_1166_not__psubset__empty,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_1167_not__psubset__empty,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_1168_psubsetD,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,C: extended_enat] :
      ( ( ord_le2529575680413868914d_enat @ A2 @ B )
     => ( ( member_Extended_enat @ C @ A2 )
       => ( member_Extended_enat @ C @ B ) ) ) ).

% psubsetD
thf(fact_1169_psubsetD,axiom,
    ! [A2: set_real,B: set_real,C: real] :
      ( ( ord_less_set_real @ A2 @ B )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B ) ) ) ).

% psubsetD
thf(fact_1170_psubsetD,axiom,
    ! [A2: set_set_nat,B: set_set_nat,C: set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B )
     => ( ( member_set_nat @ C @ A2 )
       => ( member_set_nat @ C @ B ) ) ) ).

% psubsetD
thf(fact_1171_psubsetD,axiom,
    ! [A2: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A2 @ B )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B ) ) ) ).

% psubsetD
thf(fact_1172_psubsetD,axiom,
    ! [A2: set_int,B: set_int,C: int] :
      ( ( ord_less_set_int @ A2 @ B )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B ) ) ) ).

% psubsetD
thf(fact_1173_infinite__growing,axiom,
    ! [X8: set_nat] :
      ( ( X8 != bot_bot_set_nat )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ X8 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X8 )
                & ( ord_less_nat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X8 ) ) ) ).

% infinite_growing
thf(fact_1174_infinite__growing,axiom,
    ! [X8: set_Extended_enat] :
      ( ( X8 != bot_bo7653980558646680370d_enat )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ X8 )
           => ? [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ X8 )
                & ( ord_le72135733267957522d_enat @ X5 @ Xa ) ) )
       => ~ ( finite4001608067531595151d_enat @ X8 ) ) ) ).

% infinite_growing
thf(fact_1175_infinite__growing,axiom,
    ! [X8: set_real] :
      ( ( X8 != bot_bot_set_real )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ X8 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X8 )
                & ( ord_less_real @ X5 @ Xa ) ) )
       => ~ ( finite_finite_real @ X8 ) ) ) ).

% infinite_growing
thf(fact_1176_infinite__growing,axiom,
    ! [X8: set_int] :
      ( ( X8 != bot_bot_set_int )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ X8 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X8 )
                & ( ord_less_int @ X5 @ Xa ) ) )
       => ~ ( finite_finite_int @ X8 ) ) ) ).

% infinite_growing
thf(fact_1177_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( ? [X4: set_nat] : ( member_set_nat @ X4 @ A2 ) )
      = ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_1178_ex__in__conv,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ? [X4: extended_enat] : ( member_Extended_enat @ X4 @ A2 ) )
      = ( A2 != bot_bo7653980558646680370d_enat ) ) ).

% ex_in_conv
thf(fact_1179_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ( ? [X4: real] : ( member_real @ X4 @ A2 ) )
      = ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_1180_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
      = ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_1181_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ? [X4: int] : ( member_int @ X4 @ A2 ) )
      = ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_1182_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y3: set_nat] :
          ~ ( member_set_nat @ Y3 @ A2 )
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_1183_equals0I,axiom,
    ! [A2: set_Extended_enat] :
      ( ! [Y3: extended_enat] :
          ~ ( member_Extended_enat @ Y3 @ A2 )
     => ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% equals0I
thf(fact_1184_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y3: real] :
          ~ ( member_real @ Y3 @ A2 )
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_1185_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A2 )
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_1186_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y3: int] :
          ~ ( member_int @ Y3 @ A2 )
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_1187_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_1188_equals0D,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( A2 = bot_bo7653980558646680370d_enat )
     => ~ ( member_Extended_enat @ A @ A2 ) ) ).

% equals0D
thf(fact_1189_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ ( member_real @ A @ A2 ) ) ).

% equals0D
thf(fact_1190_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_1191_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ ( member_int @ A @ A2 ) ) ).

% equals0D
thf(fact_1192_emptyE,axiom,
    ! [A: set_nat] :
      ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_1193_emptyE,axiom,
    ! [A: extended_enat] :
      ~ ( member_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ).

% emptyE
thf(fact_1194_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_1195_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_1196_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_1197_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A @ bot_bo7653980558646680370d_enat )
     => ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_uniqueI
thf(fact_1198_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_1199_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_1200_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_1201_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1202_bot_Oextremum__unique,axiom,
    ! [A: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A @ bot_bo7653980558646680370d_enat )
      = ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_unique
thf(fact_1203_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_1204_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_1205_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_1206_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1207_bot_Oextremum,axiom,
    ! [A: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ bot_bo7653980558646680370d_enat @ A ) ).

% bot.extremum
thf(fact_1208_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_1209_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_1210_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_1211_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_1212_bot_Onot__eq__extremum,axiom,
    ! [A: set_Extended_enat] :
      ( ( A != bot_bo7653980558646680370d_enat )
      = ( ord_le2529575680413868914d_enat @ bot_bo7653980558646680370d_enat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1213_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_1214_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_1215_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_1216_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1217_bot_Onot__eq__extremum,axiom,
    ! [A: extended_enat] :
      ( ( A != bot_bo4199563552545308370d_enat )
      = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1218_bot_Oextremum__strict,axiom,
    ! [A: set_Extended_enat] :
      ~ ( ord_le2529575680413868914d_enat @ A @ bot_bo7653980558646680370d_enat ) ).

% bot.extremum_strict
thf(fact_1219_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_1220_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_1221_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_1222_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1223_bot_Oextremum__strict,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).

% bot.extremum_strict
thf(fact_1224_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs2: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs2 )
          = A2 ) ) ).

% finite_list
thf(fact_1225_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs2: list_nat] :
          ( ( set_nat2 @ Xs2 )
          = A2 ) ) ).

% finite_list
thf(fact_1226_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs2: list_complex] :
          ( ( set_complex2 @ Xs2 )
          = A2 ) ) ).

% finite_list
thf(fact_1227_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs2: list_int] :
          ( ( set_int2 @ Xs2 )
          = A2 ) ) ).

% finite_list
thf(fact_1228_finite__list,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ? [Xs2: list_Extended_enat] :
          ( ( set_Extended_enat2 @ Xs2 )
          = A2 ) ) ).

% finite_list
thf(fact_1229_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_1230_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_1231_linordered__field__no__ub,axiom,
    ! [X2: real] :
    ? [X_12: real] : ( ord_less_real @ X2 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_1232_linordered__field__no__lb,axiom,
    ! [X2: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).

% linordered_field_no_lb
thf(fact_1233_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_set_nat @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_1234_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_set_int @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_1235_subset__psubset__trans,axiom,
    ! [A2: set_nat,B: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( ord_less_set_nat @ B @ C4 )
       => ( ord_less_set_nat @ A2 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_1236_subset__psubset__trans,axiom,
    ! [A2: set_int,B: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B )
     => ( ( ord_less_set_int @ B @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_1237_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B5 )
          & ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_1238_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B5 )
          & ~ ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_1239_psubset__subset__trans,axiom,
    ! [A2: set_nat,B: set_nat,C4: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B )
     => ( ( ord_less_eq_set_nat @ B @ C4 )
       => ( ord_less_set_nat @ A2 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_1240_psubset__subset__trans,axiom,
    ! [A2: set_int,B: set_int,C4: set_int] :
      ( ( ord_less_set_int @ A2 @ B )
     => ( ( ord_less_eq_set_int @ B @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_1241_psubset__imp__subset,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B )
     => ( ord_less_eq_set_nat @ A2 @ B ) ) ).

% psubset_imp_subset
thf(fact_1242_psubset__imp__subset,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( ord_less_set_int @ A2 @ B )
     => ( ord_less_eq_set_int @ A2 @ B ) ) ).

% psubset_imp_subset
thf(fact_1243_Collect__mono__iff,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
      = ( ! [X4: real] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1244_Collect__mono__iff,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
      = ( ! [X4: list_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1245_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X4: set_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1246_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1247_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X4: int] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1248_set__eq__subset,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B5 )
          & ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_1249_set__eq__subset,axiom,
    ( ( ^ [Y4: set_int,Z2: set_int] : Y4 = Z2 )
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B5 )
          & ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_1250_subset__trans,axiom,
    ! [A2: set_nat,B: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( ord_less_eq_set_nat @ B @ C4 )
       => ( ord_less_eq_set_nat @ A2 @ C4 ) ) ) ).

% subset_trans
thf(fact_1251_subset__trans,axiom,
    ! [A2: set_int,B: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B )
     => ( ( ord_less_eq_set_int @ B @ C4 )
       => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).

% subset_trans
thf(fact_1252_Collect__mono,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_mono
thf(fact_1253_Collect__mono,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1254_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1255_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1256_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_1257_subset__refl,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% subset_refl
thf(fact_1258_subset__refl,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).

% subset_refl
thf(fact_1259_subset__iff,axiom,
    ( ord_le7203529160286727270d_enat
    = ( ^ [A5: set_Extended_enat,B5: set_Extended_enat] :
        ! [T2: extended_enat] :
          ( ( member_Extended_enat @ T2 @ A5 )
         => ( member_Extended_enat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1260_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A5 )
         => ( member_real @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1261_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
        ! [T2: set_nat] :
          ( ( member_set_nat @ T2 @ A5 )
         => ( member_set_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1262_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A5 )
         => ( member_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1263_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A5 )
         => ( member_int @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1264_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% psubset_eq
thf(fact_1265_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% psubset_eq
thf(fact_1266_equalityD2,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( A2 = B )
     => ( ord_less_eq_set_nat @ B @ A2 ) ) ).

% equalityD2
thf(fact_1267_equalityD2,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
     => ( ord_less_eq_set_int @ B @ A2 ) ) ).

% equalityD2
thf(fact_1268_equalityD1,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( A2 = B )
     => ( ord_less_eq_set_nat @ A2 @ B ) ) ).

% equalityD1
thf(fact_1269_equalityD1,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
     => ( ord_less_eq_set_int @ A2 @ B ) ) ).

% equalityD1
thf(fact_1270_subset__eq,axiom,
    ( ord_le7203529160286727270d_enat
    = ( ^ [A5: set_Extended_enat,B5: set_Extended_enat] :
        ! [X4: extended_enat] :
          ( ( member_Extended_enat @ X4 @ A5 )
         => ( member_Extended_enat @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1271_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
        ! [X4: real] :
          ( ( member_real @ X4 @ A5 )
         => ( member_real @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1272_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
        ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ A5 )
         => ( member_set_nat @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1273_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A5 )
         => ( member_nat @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1274_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
        ! [X4: int] :
          ( ( member_int @ X4 @ A5 )
         => ( member_int @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1275_equalityE,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( A2 = B )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B )
         => ~ ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).

% equalityE
thf(fact_1276_equalityE,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( A2 = B )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B )
         => ~ ( ord_less_eq_set_int @ B @ A2 ) ) ) ).

% equalityE
thf(fact_1277_psubsetE,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B )
         => ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).

% psubsetE
thf(fact_1278_psubsetE,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( ord_less_set_int @ A2 @ B )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B )
         => ( ord_less_eq_set_int @ B @ A2 ) ) ) ).

% psubsetE
thf(fact_1279_subsetD,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,C: extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ B )
     => ( ( member_Extended_enat @ C @ A2 )
       => ( member_Extended_enat @ C @ B ) ) ) ).

% subsetD
thf(fact_1280_subsetD,axiom,
    ! [A2: set_real,B: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B ) ) ) ).

% subsetD
thf(fact_1281_subsetD,axiom,
    ! [A2: set_set_nat,B: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B )
     => ( ( member_set_nat @ C @ A2 )
       => ( member_set_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_1282_subsetD,axiom,
    ! [A2: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_1283_subsetD,axiom,
    ! [A2: set_int,B: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B ) ) ) ).

% subsetD
thf(fact_1284_in__mono,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,X: extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ B )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( member_Extended_enat @ X @ B ) ) ) ).

% in_mono
thf(fact_1285_in__mono,axiom,
    ! [A2: set_real,B: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A2 @ B )
     => ( ( member_real @ X @ A2 )
       => ( member_real @ X @ B ) ) ) ).

% in_mono
thf(fact_1286_in__mono,axiom,
    ! [A2: set_set_nat,B: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B )
     => ( ( member_set_nat @ X @ A2 )
       => ( member_set_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_1287_in__mono,axiom,
    ! [A2: set_nat,B: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( member_nat @ X @ A2 )
       => ( member_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_1288_in__mono,axiom,
    ! [A2: set_int,B: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ B )
     => ( ( member_int @ X @ A2 )
       => ( member_int @ X @ B ) ) ) ).

% in_mono
thf(fact_1289_finite__has__maximal,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1290_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1291_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1292_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A2 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1293_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1294_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1295_finite__has__minimal,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1296_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1297_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1298_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A2 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1299_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1300_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1301_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N5 )
         => ( ord_less_eq_nat @ X4 @ M ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1302_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M: nat] :
          ? [N: nat] :
            ( ( ord_less_eq_nat @ M @ N )
            & ( member_nat @ N @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_1303_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N5 )
         => ( ord_less_nat @ X4 @ M ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_1304_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M: nat] :
          ? [N: nat] :
            ( ( ord_less_nat @ M @ N )
            & ( member_nat @ N @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_1305_bounded__nat__set__is__finite,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ N6 )
         => ( ord_less_nat @ X5 @ N2 ) )
     => ( finite_finite_nat @ N6 ) ) ).

% bounded_nat_set_is_finite
thf(fact_1306_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M3: nat] :
          ( ( ord_less_nat @ K @ M3 )
         => ? [N7: nat] :
              ( ( ord_less_nat @ M3 @ N7 )
              & ( member_nat @ N7 @ S2 ) ) )
     => ~ ( finite_finite_nat @ S2 ) ) ).

% unbounded_k_infinite
thf(fact_1307_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [A6: set_nat] :
            ( ( finite_finite_nat @ A6 )
           => ( ! [B6: set_nat] :
                  ( ( ord_less_set_nat @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_1308_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [A6: set_complex] :
            ( ( finite3207457112153483333omplex @ A6 )
           => ( ! [B6: set_complex] :
                  ( ( ord_less_set_complex @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_1309_finite__psubset__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ! [A6: set_int] :
            ( ( finite_finite_int @ A6 )
           => ( ! [B6: set_int] :
                  ( ( ord_less_set_int @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_1310_finite__psubset__induct,axiom,
    ! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [A6: set_Extended_enat] :
            ( ( finite4001608067531595151d_enat @ A6 )
           => ( ! [B6: set_Extended_enat] :
                  ( ( ord_le2529575680413868914d_enat @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_1311_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( ( member_complex @ X2 @ S2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1312_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ~ ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ S2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( F @ ( lattic3845382081240766429at_nat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1313_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( ( member_real @ X2 @ S2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( F @ ( lattic5055836439445974935al_nat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1314_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ S2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1315_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( ( member_int @ X2 @ S2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1316_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( ( member_complex @ X2 @ S2 )
              & ( ord_le72135733267957522d_enat @ ( F @ X2 ) @ ( F @ ( lattic7796887085614042845d_enat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1317_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ~ ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ S2 )
              & ( ord_le72135733267957522d_enat @ ( F @ X2 ) @ ( F @ ( lattic1996716550891908761d_enat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1318_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: real > extended_enat] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( ( member_real @ X2 @ S2 )
              & ( ord_le72135733267957522d_enat @ ( F @ X2 ) @ ( F @ ( lattic9066027731366277983d_enat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1319_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ S2 )
              & ( ord_le72135733267957522d_enat @ ( F @ X2 ) @ ( F @ ( lattic8926238025367240251d_enat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1320_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: int > extended_enat] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( ( member_int @ X2 @ S2 )
              & ( ord_le72135733267957522d_enat @ ( F @ X2 ) @ ( F @ ( lattic6042659972569420511d_enat @ F @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1321_bot__set__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).

% bot_set_def
thf(fact_1322_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_1323_bot__set__def,axiom,
    ( bot_bo7653980558646680370d_enat
    = ( collec4429806609662206161d_enat @ bot_bo1954855461789132331enat_o ) ) ).

% bot_set_def
thf(fact_1324_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_1325_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_1326_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_1327_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1328_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M7 )
     => ? [N3: nat] :
        ! [X2: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X2 @ M7 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X2 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_1329_finite__maxlen,axiom,
    ! [M7: set_list_int] :
      ( ( finite3922522038869484883st_int @ M7 )
     => ? [N3: nat] :
        ! [X2: list_int] :
          ( ( member_list_int @ X2 @ M7 )
         => ( ord_less_nat @ ( size_size_list_int @ X2 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_1330_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M7 )
     => ? [N3: nat] :
        ! [X2: list_nat] :
          ( ( member_list_nat @ X2 @ M7 )
         => ( ord_less_nat @ ( size_size_list_nat @ X2 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_1331_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M7: nat] :
      ( ( P @ X )
     => ( ! [X5: nat] :
            ( ( P @ X5 )
           => ( ord_less_eq_nat @ X5 @ M7 ) )
       => ~ ! [M3: nat] :
              ( ( P @ M3 )
             => ~ ! [X2: nat] :
                    ( ( P @ X2 )
                   => ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1332_finite__has__maximal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ A @ A2 )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
            & ( ord_le2932123472753598470d_enat @ A @ X5 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1333_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A2 )
            & ( ord_less_eq_real @ A @ X5 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1334_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ A @ A2 )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A2 )
            & ( ord_less_eq_set_nat @ A @ X5 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1335_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A2 )
            & ( ord_less_eq_set_int @ A @ X5 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1336_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
            & ( ord_less_eq_nat @ A @ X5 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1337_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A2 )
            & ( ord_less_eq_int @ A @ X5 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1338_finite__has__minimal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ A @ A2 )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
            & ( ord_le2932123472753598470d_enat @ X5 @ A )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1339_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A2 )
            & ( ord_less_eq_real @ X5 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1340_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ A @ A2 )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A2 )
            & ( ord_less_eq_set_nat @ X5 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1341_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A2 )
            & ( ord_less_eq_set_int @ X5 @ A )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1342_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
            & ( ord_less_eq_nat @ X5 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1343_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A2 )
            & ( ord_less_eq_int @ X5 @ A )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1344_finite_OemptyI,axiom,
    finite3207457112153483333omplex @ bot_bot_set_complex ).

% finite.emptyI
thf(fact_1345_finite_OemptyI,axiom,
    finite4001608067531595151d_enat @ bot_bo7653980558646680370d_enat ).

% finite.emptyI
thf(fact_1346_finite_OemptyI,axiom,
    finite_finite_real @ bot_bot_set_real ).

% finite.emptyI
thf(fact_1347_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_1348_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_1349_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S2 )
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
thf(fact_1350_infinite__imp__nonempty,axiom,
    ! [S2: set_Extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ S2 )
     => ( S2 != bot_bo7653980558646680370d_enat ) ) ).

% infinite_imp_nonempty
thf(fact_1351_infinite__imp__nonempty,axiom,
    ! [S2: set_real] :
      ( ~ ( finite_finite_real @ S2 )
     => ( S2 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
thf(fact_1352_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_1353_infinite__imp__nonempty,axiom,
    ! [S2: set_int] :
      ( ~ ( finite_finite_int @ S2 )
     => ( S2 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_1354_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: set_nat > set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ! [X5: set_nat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: set_nat,Y3: set_nat,Z: set_nat] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z )
               => ( R @ X5 @ Z ) ) )
         => ( ! [X5: set_nat] :
                ( ( member_set_nat @ X5 @ A2 )
               => ? [Y6: set_nat] :
                    ( ( member_set_nat @ Y6 @ A2 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1355_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: complex > complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: complex,Y3: complex,Z: complex] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z )
               => ( R @ X5 @ Z ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ A2 )
               => ? [Y6: complex] :
                    ( ( member_complex @ Y6 @ A2 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1356_finite__transitivity__chain,axiom,
    ! [A2: set_Extended_enat,R: extended_enat > extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: extended_enat,Y3: extended_enat,Z: extended_enat] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z )
               => ( R @ X5 @ Z ) ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ A2 )
               => ? [Y6: extended_enat] :
                    ( ( member_Extended_enat @ Y6 @ A2 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A2 = bot_bo7653980558646680370d_enat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1357_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: real > real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X5: real] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: real,Y3: real,Z: real] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z )
               => ( R @ X5 @ Z ) ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ A2 )
               => ? [Y6: real] :
                    ( ( member_real @ Y6 @ A2 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1358_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X5: nat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y3: nat,Z: nat] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z )
               => ( R @ X5 @ Z ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ A2 )
               => ? [Y6: nat] :
                    ( ( member_nat @ Y6 @ A2 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1359_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: int > int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: int,Y3: int,Z: int] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z )
               => ( R @ X5 @ Z ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ A2 )
               => ? [Y6: int] :
                    ( ( member_int @ Y6 @ A2 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1360_finite__subset,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B )
     => ( ( finite3207457112153483333omplex @ B )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_subset
thf(fact_1361_finite__subset,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ B )
     => ( ( finite4001608067531595151d_enat @ B )
       => ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% finite_subset
thf(fact_1362_finite__subset,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( finite_finite_nat @ B )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_1363_finite__subset,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B )
     => ( ( finite_finite_int @ B )
       => ( finite_finite_int @ A2 ) ) ) ).

% finite_subset
thf(fact_1364_infinite__super,axiom,
    ! [S2: set_complex,T3: set_complex] :
      ( ( ord_le211207098394363844omplex @ S2 @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S2 )
       => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).

% infinite_super
thf(fact_1365_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_1366_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_1367_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_1368_rev__finite__subset,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1369_rev__finite__subset,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1370_rev__finite__subset,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1371_rev__finite__subset,axiom,
    ! [B: set_int,A2: set_int] :
      ( ( finite_finite_int @ B )
     => ( ( ord_less_eq_set_int @ A2 @ B )
       => ( finite_finite_int @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1372_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( S2 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S2 )
         => ( ord_less_eq_real @ ( F @ ( lattic8794016678065449205x_real @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1373_arg__min__least,axiom,
    ! [S2: set_Extended_enat,Y: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ( ( member_Extended_enat @ Y @ S2 )
         => ( ord_less_eq_real @ ( F @ ( lattic1189837152898106425t_real @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1374_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ( ( member_real @ Y @ S2 )
         => ( ord_less_eq_real @ ( F @ ( lattic8440615504127631091l_real @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1375_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: nat > real] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S2 )
         => ( ord_less_eq_real @ ( F @ ( lattic488527866317076247t_real @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1376_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ( ( member_int @ Y @ S2 )
         => ( ord_less_eq_real @ ( F @ ( lattic2675449441010098035t_real @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1377_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( S2 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S2 )
         => ( ord_less_eq_nat @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1378_arg__min__least,axiom,
    ! [S2: set_Extended_enat,Y: extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ( ( member_Extended_enat @ Y @ S2 )
         => ( ord_less_eq_nat @ ( F @ ( lattic3845382081240766429at_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1379_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ( ( member_real @ Y @ S2 )
         => ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1380_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S2 )
         => ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1381_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ( ( member_int @ Y @ S2 )
         => ( ord_less_eq_nat @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1382_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( P @ A4 @ B4 )
          = ( P @ B4 @ A4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
       => ( ! [A4: nat,B4: nat] :
              ( ( P @ A4 @ B4 )
             => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
         => ( P @ A @ B2 ) ) ) ) ).

% Euclid_induct
thf(fact_1383_nat__descend__induct,axiom,
    ! [N2: nat,P: nat > $o,M2: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N2 @ K3 )
         => ( P @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
           => ( ! [I5: nat] :
                  ( ( ord_less_nat @ K3 @ I5 )
                 => ( P @ I5 ) )
             => ( P @ K3 ) ) )
       => ( P @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_1384_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X5: set_nat] :
          ~ ( member_set_nat @ X5 @ A2 )
     => ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_1385_subset__emptyI,axiom,
    ! [A2: set_Extended_enat] :
      ( ! [X5: extended_enat] :
          ~ ( member_Extended_enat @ X5 @ A2 )
     => ( ord_le7203529160286727270d_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ).

% subset_emptyI
thf(fact_1386_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X5: real] :
          ~ ( member_real @ X5 @ A2 )
     => ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_1387_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X5: nat] :
          ~ ( member_nat @ X5 @ A2 )
     => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_1388_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X5: int] :
          ~ ( member_int @ X5 @ A2 )
     => ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_1389_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X
           != ( suc @ N3 ) ) ) ).

% list_decode.cases
thf(fact_1390_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_1391_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N3: nat] :
            ( ~ ( P @ N3 )
            & ( P @ ( suc @ N3 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1392_add__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ B2 @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_1393_add__0__iff,axiom,
    ! [B2: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ B2 @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_1394_add__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ B2 @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_1395_add__0__iff,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2
        = ( plus_plus_complex @ B2 @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_1396_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1397_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_1398_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_1399_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_1400_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_1401_less__numeral__extra_I3_J,axiom,
    ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ) ).

% less_numeral_extra(3)
thf(fact_1402_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_1403_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_1404_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_1405_verit__la__disequality,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_real @ A @ B2 )
      | ~ ( ord_less_eq_real @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_1406_verit__la__disequality,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_nat @ A @ B2 )
      | ~ ( ord_less_eq_nat @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_1407_verit__la__disequality,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_int @ A @ B2 )
      | ~ ( ord_less_eq_int @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_1408_verit__comp__simplify1_I2_J,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1409_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1410_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1411_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1412_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1413_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1414_verit__comp__simplify1_I1_J,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1415_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1416_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1417_is__num__normalize_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1418_is__num__normalize_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1419_le__numeral__extra_I3_J,axiom,
    ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).

% le_numeral_extra(3)
thf(fact_1420_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_1421_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1422_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_1423_verit__comp__simplify1_I3_J,axiom,
    ! [B7: extended_enat,A7: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ B7 @ A7 ) )
      = ( ord_le72135733267957522d_enat @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1424_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_1425_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_1426_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_1427_Collect__empty__eq__bot,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( P = bot_bot_list_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1428_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_1429_Collect__empty__eq__bot,axiom,
    ! [P: extended_enat > $o] :
      ( ( ( collec4429806609662206161d_enat @ P )
        = bot_bo7653980558646680370d_enat )
      = ( P = bot_bo1954855461789132331enat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1430_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_1431_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_1432_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_1433_bot__empty__eq,axiom,
    ( bot_bot_set_nat_o
    = ( ^ [X4: set_nat] : ( member_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1434_bot__empty__eq,axiom,
    ( bot_bo1954855461789132331enat_o
    = ( ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) ).

% bot_empty_eq
thf(fact_1435_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X4: real] : ( member_real @ X4 @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_1436_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1437_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X4: int] : ( member_int @ X4 @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_1438_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_1439_complete__interval,axiom,
    ! [A: extended_enat,B2: extended_enat,P: extended_enat > $o] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( P @ A )
       => ( ~ ( P @ B2 )
         => ? [C2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ A @ C2 )
              & ( ord_le2932123472753598470d_enat @ C2 @ B2 )
              & ! [X2: extended_enat] :
                  ( ( ( ord_le2932123472753598470d_enat @ A @ X2 )
                    & ( ord_le72135733267957522d_enat @ X2 @ C2 ) )
                 => ( P @ X2 ) )
              & ! [D3: extended_enat] :
                  ( ! [X5: extended_enat] :
                      ( ( ( ord_le2932123472753598470d_enat @ A @ X5 )
                        & ( ord_le72135733267957522d_enat @ X5 @ D3 ) )
                     => ( P @ X5 ) )
                 => ( ord_le2932123472753598470d_enat @ D3 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1440_complete__interval,axiom,
    ! [A: real,B2: real,P: real > $o] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( P @ A )
       => ( ~ ( P @ B2 )
         => ? [C2: real] :
              ( ( ord_less_eq_real @ A @ C2 )
              & ( ord_less_eq_real @ C2 @ B2 )
              & ! [X2: real] :
                  ( ( ( ord_less_eq_real @ A @ X2 )
                    & ( ord_less_real @ X2 @ C2 ) )
                 => ( P @ X2 ) )
              & ! [D3: real] :
                  ( ! [X5: real] :
                      ( ( ( ord_less_eq_real @ A @ X5 )
                        & ( ord_less_real @ X5 @ D3 ) )
                     => ( P @ X5 ) )
                 => ( ord_less_eq_real @ D3 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1441_complete__interval,axiom,
    ! [A: nat,B2: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( P @ A )
       => ( ~ ( P @ B2 )
         => ? [C2: nat] :
              ( ( ord_less_eq_nat @ A @ C2 )
              & ( ord_less_eq_nat @ C2 @ B2 )
              & ! [X2: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X2 )
                    & ( ord_less_nat @ X2 @ C2 ) )
                 => ( P @ X2 ) )
              & ! [D3: nat] :
                  ( ! [X5: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X5 )
                        & ( ord_less_nat @ X5 @ D3 ) )
                     => ( P @ X5 ) )
                 => ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1442_complete__interval,axiom,
    ! [A: int,B2: int,P: int > $o] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( P @ A )
       => ( ~ ( P @ B2 )
         => ? [C2: int] :
              ( ( ord_less_eq_int @ A @ C2 )
              & ( ord_less_eq_int @ C2 @ B2 )
              & ! [X2: int] :
                  ( ( ( ord_less_eq_int @ A @ X2 )
                    & ( ord_less_int @ X2 @ C2 ) )
                 => ( P @ X2 ) )
              & ! [D3: int] :
                  ( ! [X5: int] :
                      ( ( ( ord_less_eq_int @ A @ X5 )
                        & ( ord_less_int @ X5 @ D3 ) )
                     => ( P @ X5 ) )
                 => ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1443_pinf_I6_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ X2 )
     => ~ ( ord_le2932123472753598470d_enat @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1444_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ~ ( ord_less_eq_real @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1445_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1446_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ~ ( ord_less_eq_int @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1447_pinf_I8_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ X2 )
     => ( ord_le2932123472753598470d_enat @ T @ X2 ) ) ).

% pinf(8)
thf(fact_1448_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( ord_less_eq_real @ T @ X2 ) ) ).

% pinf(8)
thf(fact_1449_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ord_less_eq_nat @ T @ X2 ) ) ).

% pinf(8)
thf(fact_1450_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( ord_less_eq_int @ T @ X2 ) ) ).

% pinf(8)
thf(fact_1451_minf_I6_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Z )
     => ( ord_le2932123472753598470d_enat @ X2 @ T ) ) ).

% minf(6)
thf(fact_1452_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( ord_less_eq_real @ X2 @ T ) ) ).

% minf(6)
thf(fact_1453_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ord_less_eq_nat @ X2 @ T ) ) ).

% minf(6)
thf(fact_1454_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ord_less_eq_int @ X2 @ T ) ) ).

% minf(6)
thf(fact_1455_minf_I8_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Z )
     => ~ ( ord_le2932123472753598470d_enat @ T @ X2 ) ) ).

% minf(8)
thf(fact_1456_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ~ ( ord_less_eq_real @ T @ X2 ) ) ).

% minf(8)
thf(fact_1457_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).

% minf(8)
thf(fact_1458_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ~ ( ord_less_eq_int @ T @ X2 ) ) ).

% minf(8)
thf(fact_1459_count__notin,axiom,
    ! [X: extended_enat,Xs: list_Extended_enat] :
      ( ~ ( member_Extended_enat @ X @ ( set_Extended_enat2 @ Xs ) )
     => ( ( count_101369445342291426d_enat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1460_count__notin,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
     => ( ( count_list_real @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1461_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_1462_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_1463_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_1464_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_1465__C10_C,axiom,
    ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ xa @ ( ord_max_nat @ mi @ ma ) ) ) @ deg @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) ) @ ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ mi @ na ) ) @ summary ) ) ) ).

% "10"
thf(fact_1466_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( Deg = N2 ) ) ).

% deg_deg_n
thf(fact_1467_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S3 ) ) ) ).

% deg_SUcn_Node
thf(fact_1468_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
             => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_1469_max__bot2,axiom,
    ! [X: set_Extended_enat] :
      ( ( ord_ma4205026669011143323d_enat @ X @ bot_bo7653980558646680370d_enat )
      = X ) ).

% max_bot2
thf(fact_1470_max__bot2,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ X @ bot_bot_set_real )
      = X ) ).

% max_bot2
thf(fact_1471_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_1472_max__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ X @ bot_bot_set_int )
      = X ) ).

% max_bot2
thf(fact_1473_max__bot2,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ X @ bot_bot_nat )
      = X ) ).

% max_bot2
thf(fact_1474_max__bot,axiom,
    ! [X: set_Extended_enat] :
      ( ( ord_ma4205026669011143323d_enat @ bot_bo7653980558646680370d_enat @ X )
      = X ) ).

% max_bot
thf(fact_1475_max__bot,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ bot_bot_set_real @ X )
      = X ) ).

% max_bot
thf(fact_1476_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_1477_max__bot,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X )
      = X ) ).

% max_bot
thf(fact_1478_max__bot,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X )
      = X ) ).

% max_bot
thf(fact_1479_max__Suc__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ M2 @ N2 ) ) ) ).

% max_Suc_Suc
thf(fact_1480_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ord_max_nat @ A @ B2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_1481_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_1482_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B2 ) )
      = ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_1483_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_1484_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_1485_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_1486_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_1487_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_1488_max__def,axiom,
    ( ord_max_real
    = ( ^ [A3: real,B3: real] : ( if_real @ ( ord_less_eq_real @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1489_max__def,axiom,
    ( ord_max_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1490_max__def,axiom,
    ( ord_max_set_int
    = ( ^ [A3: set_int,B3: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1491_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1492_max__def,axiom,
    ( ord_max_int
    = ( ^ [A3: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1493_max__absorb1,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ( ord_max_real @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_1494_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_max_set_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_1495_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_1496_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_1497_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_max_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_1498_max__absorb2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_max_real @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_1499_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_max_set_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_1500_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_1501_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_1502_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_max_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_1503_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z3 )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z3 ) @ ( plus_plus_real @ Y @ Z3 ) ) ) ).

% max_add_distrib_left
thf(fact_1504_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z3 )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z3 ) @ ( plus_plus_nat @ Y @ Z3 ) ) ) ).

% max_add_distrib_left
thf(fact_1505_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z3 )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z3 ) @ ( plus_plus_int @ Y @ Z3 ) ) ) ).

% max_add_distrib_left
thf(fact_1506_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z3 ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z3 ) ) ) ).

% max_add_distrib_right
thf(fact_1507_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z3 ) ) ) ).

% max_add_distrib_right
thf(fact_1508_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z3 ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z3 ) ) ) ).

% max_add_distrib_right
thf(fact_1509_nat__add__max__left,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N2 ) @ Q3 )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q3 ) @ ( plus_plus_nat @ N2 @ Q3 ) ) ) ).

% nat_add_max_left
thf(fact_1510_nat__add__max__right,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N2 @ Q3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ).

% nat_add_max_right
thf(fact_1511_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_1512_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ~ ( ord_less_nat @ T @ X2 ) ) ).

% minf(7)
thf(fact_1513_minf_I7_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Z )
     => ~ ( ord_le72135733267957522d_enat @ T @ X2 ) ) ).

% minf(7)
thf(fact_1514_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ~ ( ord_less_real @ T @ X2 ) ) ).

% minf(7)
thf(fact_1515_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ~ ( ord_less_int @ T @ X2 ) ) ).

% minf(7)
thf(fact_1516_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ord_less_nat @ X2 @ T ) ) ).

% minf(5)
thf(fact_1517_minf_I5_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Z )
     => ( ord_le72135733267957522d_enat @ X2 @ T ) ) ).

% minf(5)
thf(fact_1518_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( ord_less_real @ X2 @ T ) ) ).

% minf(5)
thf(fact_1519_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ord_less_int @ X2 @ T ) ) ).

% minf(5)
thf(fact_1520_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1521_minf_I4_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1522_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1523_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1524_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1525_minf_I3_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1526_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1527_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1528_minf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z4 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q4 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1529_minf_I2_J,axiom,
    ! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q4: extended_enat > $o] :
      ( ? [Z4: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X5 @ Z4 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: extended_enat] :
          ! [X2: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q4 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1530_minf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q4: real > $o] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z4 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( ( ord_less_real @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q4 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1531_minf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z4 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q4 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1532_minf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z4 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q4 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1533_minf_I1_J,axiom,
    ! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q4: extended_enat > $o] :
      ( ? [Z4: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X5 @ Z4 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: extended_enat] :
          ! [X2: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q4 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1534_minf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q4: real > $o] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z4 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( ( ord_less_real @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q4 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1535_minf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z4 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q4 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1536_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ord_less_nat @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1537_pinf_I7_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ X2 )
     => ( ord_le72135733267957522d_enat @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1538_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( ord_less_real @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1539_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( ord_less_int @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1540_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ~ ( ord_less_nat @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1541_pinf_I5_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ X2 )
     => ~ ( ord_le72135733267957522d_enat @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1542_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ~ ( ord_less_real @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1543_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ~ ( ord_less_int @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1544_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1545_pinf_I4_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1546_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1547_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1548_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1549_pinf_I3_J,axiom,
    ! [T: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1550_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1551_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1552_pinf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z4 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q4 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1553_pinf_I2_J,axiom,
    ! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q4: extended_enat > $o] :
      ( ? [Z4: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ Z4 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: extended_enat] :
          ! [X2: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q4 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1554_pinf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q4: real > $o] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z4 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( ( ord_less_real @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q4 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1555_pinf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z4 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q4 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1556_pinf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z4 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q4 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1557_pinf_I1_J,axiom,
    ! [P: extended_enat > $o,P4: extended_enat > $o,Q: extended_enat > $o,Q4: extended_enat > $o] :
      ( ? [Z4: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ Z4 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: extended_enat] :
          ! [X2: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q4 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1558_pinf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q4: real > $o] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z4 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( ( ord_less_real @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q4 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1559_pinf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z4 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q4 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1560_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B4: real] :
      ( ( ord_less_real @ A @ B4 )
      | ( ord_less_real @ B4 @ A ) ) ).

% ex_gt_or_lt
thf(fact_1561_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_1562_count__le__length,axiom,
    ! [Xs: list_int,X: int] : ( ord_less_eq_nat @ ( count_list_int @ Xs @ X ) @ ( size_size_list_int @ Xs ) ) ).

% count_le_length
thf(fact_1563_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_1564_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_1565_max_Oabsorb3,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_max_nat @ A @ B2 )
        = A ) ) ).

% max.absorb3
thf(fact_1566_max_Oabsorb3,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
        = A ) ) ).

% max.absorb3
thf(fact_1567_max_Oabsorb3,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_max_real @ A @ B2 )
        = A ) ) ).

% max.absorb3
thf(fact_1568_max_Oabsorb3,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_max_int @ A @ B2 )
        = A ) ) ).

% max.absorb3
thf(fact_1569_max_Oabsorb4,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_max_nat @ A @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_1570_max_Oabsorb4,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_1571_max_Oabsorb4,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_max_real @ A @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_1572_max_Oabsorb4,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_max_int @ A @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_1573_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z3 )
      = ( ( ord_less_nat @ X @ Z3 )
        & ( ord_less_nat @ Y @ Z3 ) ) ) ).

% max_less_iff_conj
thf(fact_1574_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y ) @ Z3 )
      = ( ( ord_le72135733267957522d_enat @ X @ Z3 )
        & ( ord_le72135733267957522d_enat @ Y @ Z3 ) ) ) ).

% max_less_iff_conj
thf(fact_1575_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z3 )
      = ( ( ord_less_real @ X @ Z3 )
        & ( ord_less_real @ Y @ Z3 ) ) ) ).

% max_less_iff_conj
thf(fact_1576_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z3 )
      = ( ( ord_less_int @ X @ Z3 )
        & ( ord_less_int @ Y @ Z3 ) ) ) ).

% max_less_iff_conj
thf(fact_1577_max_Oabsorb1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_max_real @ A @ B2 )
        = A ) ) ).

% max.absorb1
thf(fact_1578_max_Oabsorb1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_max_nat @ A @ B2 )
        = A ) ) ).

% max.absorb1
thf(fact_1579_max_Oabsorb1,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_max_int @ A @ B2 )
        = A ) ) ).

% max.absorb1
thf(fact_1580_max_Oabsorb2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_max_real @ A @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_1581_max_Oabsorb2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_max_nat @ A @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_1582_max_Oabsorb2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_max_int @ A @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_1583_max_Obounded__iff,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( ord_max_real @ B2 @ C ) @ A )
      = ( ( ord_less_eq_real @ B2 @ A )
        & ( ord_less_eq_real @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_1584_max_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A )
      = ( ( ord_less_eq_nat @ B2 @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_1585_max_Obounded__iff,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A )
      = ( ( ord_less_eq_int @ B2 @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_1586_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_1587_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_1588_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_1589_max_OcoboundedI2,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_eq_real @ C @ B2 )
     => ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_1590_max_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B2 )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_1591_max_OcoboundedI2,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( ord_less_eq_int @ C @ B2 )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_1592_max_OcoboundedI1,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ C @ A )
     => ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_1593_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_1594_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_1595_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_max_real @ A3 @ B3 )
          = B3 ) ) ) ).

% max.absorb_iff2
thf(fact_1596_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_max_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% max.absorb_iff2
thf(fact_1597_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_max_int @ A3 @ B3 )
          = B3 ) ) ) ).

% max.absorb_iff2
thf(fact_1598_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_max_real @ A3 @ B3 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_1599_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_max_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_1600_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_max_int @ A3 @ B3 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_1601_le__max__iff__disj,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ Z3 @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_eq_real @ Z3 @ X )
        | ( ord_less_eq_real @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_1602_le__max__iff__disj,axiom,
    ! [Z3: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z3 @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z3 @ X )
        | ( ord_less_eq_nat @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_1603_le__max__iff__disj,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z3 @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z3 @ X )
        | ( ord_less_eq_int @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_1604_max_Ocobounded2,axiom,
    ! [B2: real,A: real] : ( ord_less_eq_real @ B2 @ ( ord_max_real @ A @ B2 ) ) ).

% max.cobounded2
thf(fact_1605_max_Ocobounded2,axiom,
    ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( ord_max_nat @ A @ B2 ) ) ).

% max.cobounded2
thf(fact_1606_max_Ocobounded2,axiom,
    ! [B2: int,A: int] : ( ord_less_eq_int @ B2 @ ( ord_max_int @ A @ B2 ) ) ).

% max.cobounded2
thf(fact_1607_max_Ocobounded1,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ A @ ( ord_max_real @ A @ B2 ) ) ).

% max.cobounded1
thf(fact_1608_max_Ocobounded1,axiom,
    ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B2 ) ) ).

% max.cobounded1
thf(fact_1609_max_Ocobounded1,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B2 ) ) ).

% max.cobounded1
thf(fact_1610_max_Oorder__iff,axiom,
    ( ord_less_eq_real
    = ( ^ [B3: real,A3: real] :
          ( A3
          = ( ord_max_real @ A3 @ B3 ) ) ) ) ).

% max.order_iff
thf(fact_1611_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( A3
          = ( ord_max_nat @ A3 @ B3 ) ) ) ) ).

% max.order_iff
thf(fact_1612_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( A3
          = ( ord_max_int @ A3 @ B3 ) ) ) ) ).

% max.order_iff
thf(fact_1613_max_OboundedI,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ A )
       => ( ord_less_eq_real @ ( ord_max_real @ B2 @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_1614_max_OboundedI,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_1615_max_OboundedI,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_1616_max_OboundedE,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( ord_max_real @ B2 @ C ) @ A )
     => ~ ( ( ord_less_eq_real @ B2 @ A )
         => ~ ( ord_less_eq_real @ C @ A ) ) ) ).

% max.boundedE
thf(fact_1617_max_OboundedE,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B2 @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_1618_max_OboundedE,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B2 @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_1619_max_OorderI,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( ord_max_real @ A @ B2 ) )
     => ( ord_less_eq_real @ B2 @ A ) ) ).

% max.orderI
thf(fact_1620_max_OorderI,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A ) ) ).

% max.orderI
thf(fact_1621_max_OorderI,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( ord_max_int @ A @ B2 ) )
     => ( ord_less_eq_int @ B2 @ A ) ) ).

% max.orderI
thf(fact_1622_max_OorderE,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( A
        = ( ord_max_real @ A @ B2 ) ) ) ).

% max.orderE
thf(fact_1623_max_OorderE,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( A
        = ( ord_max_nat @ A @ B2 ) ) ) ).

% max.orderE
thf(fact_1624_max_OorderE,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( A
        = ( ord_max_int @ A @ B2 ) ) ) ).

% max.orderE
thf(fact_1625_max_Omono,axiom,
    ! [C: real,A: real,D: real,B2: real] :
      ( ( ord_less_eq_real @ C @ A )
     => ( ( ord_less_eq_real @ D @ B2 )
       => ( ord_less_eq_real @ ( ord_max_real @ C @ D ) @ ( ord_max_real @ A @ B2 ) ) ) ) ).

% max.mono
thf(fact_1626_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B2 )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B2 ) ) ) ) ).

% max.mono
thf(fact_1627_max_Omono,axiom,
    ! [C: int,A: int,D: int,B2: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B2 )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B2 ) ) ) ) ).

% max.mono
thf(fact_1628_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ord_less_nat @ C @ B2 )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_1629_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B2: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ B2 )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_1630_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ B2 )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_1631_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( ord_less_int @ C @ B2 )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_1632_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_1633_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ A )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_1634_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_1635_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_1636_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( A3
            = ( ord_max_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1637_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( A3
            = ( ord_ma741700101516333627d_enat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1638_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A3: real] :
          ( ( A3
            = ( ord_max_real @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1639_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( A3
            = ( ord_max_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1640_max_Ostrict__boundedE,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B2 @ C ) @ A )
     => ~ ( ( ord_less_nat @ B2 @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_1641_max_Ostrict__boundedE,axiom,
    ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A )
     => ~ ( ( ord_le72135733267957522d_enat @ B2 @ A )
         => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_1642_max_Ostrict__boundedE,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B2 @ C ) @ A )
     => ~ ( ( ord_less_real @ B2 @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_1643_max_Ostrict__boundedE,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B2 @ C ) @ A )
     => ~ ( ( ord_less_int @ B2 @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_1644_less__max__iff__disj,axiom,
    ! [Z3: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z3 @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z3 @ X )
        | ( ord_less_nat @ Z3 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_1645_less__max__iff__disj,axiom,
    ! [Z3: extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z3 @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Z3 @ X )
        | ( ord_le72135733267957522d_enat @ Z3 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_1646_less__max__iff__disj,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_real @ Z3 @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_real @ Z3 @ X )
        | ( ord_less_real @ Z3 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_1647_less__max__iff__disj,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ Z3 @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z3 @ X )
        | ( ord_less_int @ Z3 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_1648_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_1649_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1650_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1651_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_VEBT_VEBT,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs ) @ M2 )
        = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N2 @ M2 ) @ ( nth_VEBT_VEBT @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1652_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_int,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_int @ Xs ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N2 @ Xs ) @ M2 )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N2 @ M2 ) @ ( nth_int @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1653_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N2 @ Xs ) @ M2 )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N2 @ M2 ) @ ( nth_nat @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1654_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 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
            & ( X
              = ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1655_find__Some__iff2,axiom,
    ! [X: num,P: num > $o,Xs: list_num] :
      ( ( ( some_num @ X )
        = ( find_num @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_num @ Xs ) )
            & ( P @ ( nth_num @ Xs @ I3 ) )
            & ( X
              = ( nth_num @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P @ ( nth_num @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1656_find__Some__iff2,axiom,
    ! [X: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( ( some_VEBT_VEBT @ X )
        = ( find_VEBT_VEBT @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) )
            & ( X
              = ( nth_VEBT_VEBT @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1657_find__Some__iff2,axiom,
    ! [X: int,P: int > $o,Xs: list_int] :
      ( ( ( some_int @ X )
        = ( find_int @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
            & ( P @ ( nth_int @ Xs @ I3 ) )
            & ( X
              = ( nth_int @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P @ ( nth_int @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1658_find__Some__iff2,axiom,
    ! [X: nat,P: nat > $o,Xs: list_nat] :
      ( ( ( some_nat @ X )
        = ( find_nat @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I3 ) )
            & ( X
              = ( nth_nat @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1659_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 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
            & ( X
              = ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1660_find__Some__iff,axiom,
    ! [P: num > $o,Xs: list_num,X: num] :
      ( ( ( find_num @ P @ Xs )
        = ( some_num @ X ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_num @ Xs ) )
            & ( P @ ( nth_num @ Xs @ I3 ) )
            & ( X
              = ( nth_num @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P @ ( nth_num @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1661_find__Some__iff,axiom,
    ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( find_VEBT_VEBT @ P @ Xs )
        = ( some_VEBT_VEBT @ X ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) )
            & ( X
              = ( nth_VEBT_VEBT @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1662_find__Some__iff,axiom,
    ! [P: int > $o,Xs: list_int,X: int] :
      ( ( ( find_int @ P @ Xs )
        = ( some_int @ X ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
            & ( P @ ( nth_int @ Xs @ I3 ) )
            & ( X
              = ( nth_int @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P @ ( nth_int @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1663_find__Some__iff,axiom,
    ! [P: nat > $o,Xs: list_nat,X: nat] :
      ( ( ( find_nat @ P @ Xs )
        = ( some_nat @ X ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I3 ) )
            & ( X
              = ( nth_nat @ Xs @ I3 ) )
            & ! [J2: nat] :
                ( ( ord_less_nat @ J2 @ I3 )
               => ~ ( P @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1664__C7_C,axiom,
    ( ( mi != ma )
   => ! [I5: nat] :
        ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I5 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I5 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [Y6: nat] :
              ( ( ( ( vEBT_VEBT_high @ Y6 @ na )
                  = I5 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I5 ) @ ( vEBT_VEBT_low @ Y6 @ na ) ) )
             => ( ( ord_less_nat @ mi @ Y6 )
                & ( ord_less_eq_nat @ Y6 @ ma ) ) ) ) ) ) ).

% "7"
thf(fact_1665_listrel1__iff__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) )
      = ( ? [Y5: product_prod_nat_nat,N: nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ Y5 ) @ R2 )
            & ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( Ys
              = ( list_u6180841689913720943at_nat @ Xs @ N @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1666_listrel1__iff__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
      = ( ? [Y5: vEBT_VEBT,N: nat] :
            ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ Y5 ) @ R2 )
            & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( Ys
              = ( list_u1324408373059187874T_VEBT @ Xs @ N @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1667_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
      = ( ? [Y5: int,N: nat] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N ) @ Y5 ) @ R2 )
            & ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
            & ( Ys
              = ( list_update_int @ Xs @ N @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1668_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
      = ( ? [Y5: nat,N: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N ) @ Y5 ) @ R2 )
            & ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1669_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1670_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X22: num] :
      ( ( size_option_num @ X @ ( some_num @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1671_intind,axiom,
    ! [I: nat,N2: nat,P: nat > $o,X: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X )
       => ( P @ ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_1672_intind,axiom,
    ! [I: nat,N2: nat,P: int > $o,X: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X )
       => ( P @ ( nth_int @ ( replicate_int @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_1673_intind,axiom,
    ! [I: nat,N2: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_1674_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_1675_gen__length__def,axiom,
    ( gen_length_VEBT_VEBT
    = ( ^ [N: nat,Xs3: list_VEBT_VEBT] : ( plus_plus_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_1676_gen__length__def,axiom,
    ( gen_length_int
    = ( ^ [N: nat,Xs3: list_int] : ( plus_plus_nat @ N @ ( size_size_list_int @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_1677_gen__length__def,axiom,
    ( gen_length_nat
    = ( ^ [N: nat,Xs3: list_nat] : ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_1678__C2_C,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "2"
thf(fact_1679__C5_Ohyps_C_I8_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "5.hyps"(8)
thf(fact_1680__C5_Oprems_C,axiom,
    ord_less_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "5.prems"
thf(fact_1681__092_060open_062i_A_060_A2_A_094_Am_092_060close_062,axiom,
    ord_less_nat @ i @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).

% \<open>i < 2 ^ m\<close>
thf(fact_1682__C5_OIH_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X2 @ na )
        & ! [Xa: nat] :
            ( ( ord_less_nat @ Xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
           => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ X2 @ Xa ) @ na ) ) ) ) ).

% "5.IH"(1)
thf(fact_1683__C4_C,axiom,
    ! [I5: nat] :
      ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I5 ) @ X6 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I5 ) ) ) ).

% "4"
thf(fact_1684_high__bound__aux,axiom,
    ! [Ma: nat,N2: nat,M2: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M2 ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% high_bound_aux
thf(fact_1685_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N2: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N2 )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% member_bound
thf(fact_1686_numeral__le__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_eq_num @ M2 @ N2 ) ) ).

% numeral_le_iff
thf(fact_1687_numeral__le__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_num @ M2 @ N2 ) ) ).

% numeral_le_iff
thf(fact_1688_numeral__le__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_eq_num @ M2 @ N2 ) ) ).

% numeral_le_iff
thf(fact_1689_numeral__le__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_num @ M2 @ N2 ) ) ).

% numeral_le_iff
thf(fact_1690_numeral__less__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ M2 @ N2 ) ) ).

% numeral_less_iff
thf(fact_1691_numeral__less__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_num @ M2 @ N2 ) ) ).

% numeral_less_iff
thf(fact_1692_numeral__less__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ M2 @ N2 ) ) ).

% numeral_less_iff
thf(fact_1693_numeral__less__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ M2 @ N2 ) ) ).

% numeral_less_iff
thf(fact_1694_numeral__plus__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1695_numeral__plus__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1696_numeral__plus__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1697_numeral__plus__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1698_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_1699_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z3 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_1700_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_1701_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z3 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_1702_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: 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 @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_1703_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_1704_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_1705_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_1706_replicate__eq__replicate,axiom,
    ! [M2: nat,X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M2 @ X )
        = ( replicate_VEBT_VEBT @ N2 @ Y ) )
      = ( ( M2 = N2 )
        & ( ( M2 != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_1707_length__replicate,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) )
      = N2 ) ).

% length_replicate
thf(fact_1708_length__replicate,axiom,
    ! [N2: nat,X: int] :
      ( ( size_size_list_int @ ( replicate_int @ N2 @ X ) )
      = N2 ) ).

% length_replicate
thf(fact_1709_length__replicate,axiom,
    ! [N2: nat,X: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N2 @ X ) )
      = N2 ) ).

% length_replicate
thf(fact_1710_length__enumerate,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs ) )
      = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_enumerate
thf(fact_1711_length__enumerate,axiom,
    ! [N2: nat,Xs: list_int] :
      ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N2 @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_enumerate
thf(fact_1712_length__enumerate,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_1713_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( 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_1714__C5_OIH_C_I2_J,axiom,
    ! [X: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ summary @ X ) @ m ) ) ).

% "5.IH"(2)
thf(fact_1715_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_1716_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_1717_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_1718_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_1719_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_1720_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_1721_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_1722_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_1723_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_1724_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_1725_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_1726_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_1727_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_1728__C6_C,axiom,
    ( ( ord_less_eq_nat @ mi @ ma )
    & ( ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ) ) ).

% "6"
thf(fact_1729_myIHs,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X @ na )
       => ( ( ord_less_nat @ Xa2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
         => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ X @ Xa2 ) @ na ) ) ) ) ).

% myIHs
thf(fact_1730_in__set__replicate,axiom,
    ! [X: extended_enat,N2: nat,Y: extended_enat] :
      ( ( member_Extended_enat @ X @ ( set_Extended_enat2 @ ( replic7216382294607269926d_enat @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1731_in__set__replicate,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1732_in__set__replicate,axiom,
    ! [X: set_nat,N2: nat,Y: set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1733_in__set__replicate,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1734_in__set__replicate,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1735_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1736_Bex__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ? [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
            & ( P @ X4 ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_1737_Bex__set__replicate,axiom,
    ! [N2: nat,A: nat,P: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
            & ( P @ X4 ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_1738_Bex__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
            & ( P @ X4 ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_1739_Ball__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
           => ( P @ X4 ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_1740_Ball__set__replicate,axiom,
    ! [N2: nat,A: nat,P: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
           => ( P @ X4 ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_1741_Ball__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
           => ( P @ X4 ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_1742_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_1743_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_int @ ( replicate_int @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_1744_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_1745_highlowprop,axiom,
    ( ( ord_less_nat @ ( vEBT_VEBT_high @ mi @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
    & ( ord_less_nat @ ( vEBT_VEBT_low @ mi @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).

% highlowprop
thf(fact_1746_add__2__eq__Suc,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc
thf(fact_1747_add__2__eq__Suc_H,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc'
thf(fact_1748_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1749_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1750_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1751_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1752_listrel1__eq__len,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1753_listrel1__eq__len,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
     => ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1754_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1755_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_1756_find__None__iff,axiom,
    ! [P: extended_enat > $o,Xs: list_Extended_enat] :
      ( ( ( find_Extended_enat @ P @ Xs )
        = none_Extended_enat )
      = ( ~ ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff
thf(fact_1757_find__None__iff,axiom,
    ! [P: real > $o,Xs: list_real] :
      ( ( ( find_real @ P @ Xs )
        = none_real )
      = ( ~ ? [X4: real] :
              ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff
thf(fact_1758_find__None__iff,axiom,
    ! [P: set_nat > $o,Xs: list_set_nat] :
      ( ( ( find_set_nat @ P @ Xs )
        = none_set_nat )
      = ( ~ ? [X4: set_nat] :
              ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff
thf(fact_1759_find__None__iff,axiom,
    ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( ( find_VEBT_VEBT @ P @ Xs )
        = none_VEBT_VEBT )
      = ( ~ ? [X4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff
thf(fact_1760_find__None__iff,axiom,
    ! [P: int > $o,Xs: list_int] :
      ( ( ( find_int @ P @ Xs )
        = none_int )
      = ( ~ ? [X4: int] :
              ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff
thf(fact_1761_find__None__iff,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( ( find_nat @ P @ Xs )
        = none_nat )
      = ( ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff
thf(fact_1762_find__None__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
        = none_P5556105721700978146at_nat )
      = ( ~ ? [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff
thf(fact_1763_find__None__iff,axiom,
    ! [P: num > $o,Xs: list_num] :
      ( ( ( find_num @ P @ Xs )
        = none_num )
      = ( ~ ? [X4: num] :
              ( ( member_num @ X4 @ ( set_num2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff
thf(fact_1764_find__None__iff2,axiom,
    ! [P: extended_enat > $o,Xs: list_Extended_enat] :
      ( ( none_Extended_enat
        = ( find_Extended_enat @ P @ Xs ) )
      = ( ~ ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( set_Extended_enat2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff2
thf(fact_1765_find__None__iff2,axiom,
    ! [P: real > $o,Xs: list_real] :
      ( ( none_real
        = ( find_real @ P @ Xs ) )
      = ( ~ ? [X4: real] :
              ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff2
thf(fact_1766_find__None__iff2,axiom,
    ! [P: set_nat > $o,Xs: list_set_nat] :
      ( ( none_set_nat
        = ( find_set_nat @ P @ Xs ) )
      = ( ~ ? [X4: set_nat] :
              ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff2
thf(fact_1767_find__None__iff2,axiom,
    ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT
        = ( find_VEBT_VEBT @ P @ Xs ) )
      = ( ~ ? [X4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff2
thf(fact_1768_find__None__iff2,axiom,
    ! [P: int > $o,Xs: list_int] :
      ( ( none_int
        = ( find_int @ P @ Xs ) )
      = ( ~ ? [X4: int] :
              ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff2
thf(fact_1769_find__None__iff2,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( none_nat
        = ( find_nat @ P @ Xs ) )
      = ( ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff2
thf(fact_1770_find__None__iff2,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat
        = ( find_P8199882355184865565at_nat @ P @ Xs ) )
      = ( ~ ? [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff2
thf(fact_1771_find__None__iff2,axiom,
    ! [P: num > $o,Xs: list_num] :
      ( ( none_num
        = ( find_num @ P @ Xs ) )
      = ( ~ ? [X4: num] :
              ( ( member_num @ X4 @ ( set_num2 @ Xs ) )
              & ( P @ X4 ) ) ) ) ).

% find_None_iff2
thf(fact_1772_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N2: nat,M2: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M2 ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_1773_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_1774_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_1775_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N2: nat,M2: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M2 ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_1776_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_1777_less__2__cases,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_1778_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2 = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M2 ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_1779_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_1780_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).

% Suc_nat_number_of_add
thf(fact_1781_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1782_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1783_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_z5237406670263579293d_enat
     != ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1784_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1785_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1786_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F2: nat > nat > nat,A4: nat,B4: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_1787_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M2 ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_1788_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_1789_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1790_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1791_find__cong,axiom,
    ! [Xs: list_Extended_enat,Ys: list_Extended_enat,P: extended_enat > $o,Q: extended_enat > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ ( set_Extended_enat2 @ Ys ) )
           => ( ( P @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_Extended_enat @ P @ Xs )
          = ( find_Extended_enat @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1792_find__cong,axiom,
    ! [Xs: list_real,Ys: list_real,P: real > $o,Q: real > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_real2 @ Ys ) )
           => ( ( P @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_real @ P @ Xs )
          = ( find_real @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1793_find__cong,axiom,
    ! [Xs: list_set_nat,Ys: list_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: set_nat] :
            ( ( member_set_nat @ X5 @ ( set_set_nat2 @ Ys ) )
           => ( ( P @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_set_nat @ P @ Xs )
          = ( find_set_nat @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1794_find__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: vEBT_VEBT > $o,Q: vEBT_VEBT > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( P @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_VEBT_VEBT @ P @ Xs )
          = ( find_VEBT_VEBT @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1795_find__cong,axiom,
    ! [Xs: list_int,Ys: list_int,P: int > $o,Q: int > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( set_int2 @ Ys ) )
           => ( ( P @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_int @ P @ Xs )
          = ( find_int @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1796_find__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ Ys ) )
           => ( ( P @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_nat @ P @ Xs )
          = ( find_nat @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1797_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_le_zero
thf(fact_1798_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_1799_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_1800_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_1801_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1802_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_le_numeral
thf(fact_1803_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1804_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_1805_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_1806_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_less_zero
thf(fact_1807_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_1808_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_1809_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1810_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1811_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_1812_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_less_numeral
thf(fact_1813_replicate__eqI,axiom,
    ! [Xs: list_Extended_enat,N2: nat,X: extended_enat] :
      ( ( ( size_s3941691890525107288d_enat @ Xs )
        = N2 )
     => ( ! [Y3: extended_enat] :
            ( ( member_Extended_enat @ Y3 @ ( set_Extended_enat2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replic7216382294607269926d_enat @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1814_replicate__eqI,axiom,
    ! [Xs: list_real,N2: nat,X: real] :
      ( ( ( size_size_list_real @ Xs )
        = N2 )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ ( set_real2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_real @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1815_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N2: nat,X: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = N2 )
     => ( ! [Y3: set_nat] :
            ( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_set_nat @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1816_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N2: nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = N2 )
     => ( ! [Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_VEBT_VEBT @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1817_replicate__eqI,axiom,
    ! [Xs: list_int,N2: nat,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = N2 )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ ( set_int2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_int @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1818_replicate__eqI,axiom,
    ! [Xs: list_nat,N2: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = N2 )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_nat @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1819_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( X5 = X ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_1820_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
         => ( X5 = X ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_1821_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
         => ( X5 = X ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_1822_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_1823_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_1824_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2 = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M2 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I4: nat] :
                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I4 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
                                        = I4 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X5 )
                                      & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_1825_num_Osize_I5_J,axiom,
    ! [X22: num] :
      ( ( size_size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_1826_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_1827_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M2 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I4: nat] :
                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I4 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
                                        = I4 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X5 )
                                      & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_1828_length__code,axiom,
    ( size_s6755466524823107622T_VEBT
    = ( gen_length_VEBT_VEBT @ zero_zero_nat ) ) ).

% length_code
thf(fact_1829_length__code,axiom,
    ( size_size_list_int
    = ( gen_length_int @ zero_zero_nat ) ) ).

% length_code
thf(fact_1830_length__code,axiom,
    ( size_size_list_nat
    = ( gen_length_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_1831_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_1832_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_1833_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_1834_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_1835_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_1836_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_1837_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_1838_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_1839_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_1840_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_1841_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_1842_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_1843_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_1844_power__mono__iff,axiom,
    ! [A: real,B2: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B2 @ N2 ) )
            = ( ord_less_eq_real @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1845_power__mono__iff,axiom,
    ! [A: nat,B2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B2 @ N2 ) )
            = ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1846_power__mono__iff,axiom,
    ! [A: int,B2: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B2 @ N2 ) )
            = ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1847_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList2: 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 @ TreeList2 ) )
     => ( ( 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 @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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_1848_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: 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 @ TreeList2 ) )
     => ( ( 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 @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_1849_power__eq__0__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1850_power__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( power_power_real @ A @ N2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1851_power__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( ( power_power_complex @ A @ N2 )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1852_power__eq__0__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ( power_power_int @ A @ N2 )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1853_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ 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 @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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_1854_pow__sum,axiom,
    ! [A: nat,B2: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ).

% pow_sum
thf(fact_1855_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X4: nat,N: nat] : ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% high_def
thf(fact_1856__C9_C,axiom,
    ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = na ) ).

% "9"
thf(fact_1857_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_1858_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_1859_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1860_divide__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1861_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ C @ A )
        = ( divide1717551699836669952omplex @ C @ B2 ) )
      = ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1862_divide__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1863_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_1864_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1865_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1866_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1867_divide__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B2 )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
thf(fact_1868_divide__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( divide_divide_real @ A @ B2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_1869_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_1870_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1871_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1872_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1873_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ ( suc @ N2 ) )
      = zero_z5237406670263579293d_enat ) ).

% power_0_Suc
thf(fact_1874_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_1875_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_1876_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_1877_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_1878_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ ( numeral_numeral_nat @ K ) )
      = zero_z5237406670263579293d_enat ) ).

% power_zero_numeral
thf(fact_1879_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_1880_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_1881_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_1882_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_1883_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1884_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1885_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1886_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1887_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_1888_power__Suc__0,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_1889_nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_1890_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_1891_add__divide__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).

% add_divide_distrib
thf(fact_1892_divide__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).

% divide_le_0_iff
thf(fact_1893_divide__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_1894_zero__le__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_1895_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_1896_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_1897_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_1898_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_1899_divide__right__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_1900_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_1901_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_1902_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_1903_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_1904_divide__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).

% divide_less_0_iff
thf(fact_1905_divide__less__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_1906_zero__less__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_1907_divide__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_1908_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_1909_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_1910_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_1911_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_1912_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_1913_divide__le__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_1914_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z3: 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 @ Z3 )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_1915_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z3: 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 @ Z3 )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_1916_frac__le,axiom,
    ! [Y: real,X: real,W2: real,Z3: 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 @ Z3 )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_1917_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_1918_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_1919_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_1920_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_1921_power__not__zero,axiom,
    ! [A: nat,N2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N2 )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_1922_power__not__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N2 )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_1923_power__not__zero,axiom,
    ! [A: complex,N2: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N2 )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_1924_power__not__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N2 )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_1925_power__mono,axiom,
    ! [A: real,B2: real,N2: nat] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B2 @ N2 ) ) ) ) ).

% power_mono
thf(fact_1926_power__mono,axiom,
    ! [A: nat,B2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B2 @ N2 ) ) ) ) ).

% power_mono
thf(fact_1927_power__mono,axiom,
    ! [A: int,B2: int,N2: nat] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B2 @ N2 ) ) ) ) ).

% power_mono
thf(fact_1928_zero__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_1929_zero__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_1930_zero__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_1931_zero__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_1932_zero__less__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_1933_zero__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_1934_nat__power__less__imp__less,axiom,
    ! [I: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% nat_power_less_imp_less
thf(fact_1935_power__less__imp__less__base,axiom,
    ! [A: real,N2: nat,B2: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B2 @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_1936_power__less__imp__less__base,axiom,
    ! [A: nat,N2: nat,B2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B2 @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_1937_power__less__imp__less__base,axiom,
    ! [A: int,N2: nat,B2: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B2 @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_1938_power__inject__base,axiom,
    ! [A: real,N2: nat,B2: real] :
      ( ( ( power_power_real @ A @ ( suc @ N2 ) )
        = ( power_power_real @ B2 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_1939_power__inject__base,axiom,
    ! [A: nat,N2: nat,B2: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
        = ( power_power_nat @ B2 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_1940_power__inject__base,axiom,
    ! [A: int,N2: nat,B2: int] :
      ( ( ( power_power_int @ A @ ( suc @ N2 ) )
        = ( power_power_int @ B2 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_1941_power__le__imp__le__base,axiom,
    ! [A: real,N2: nat,B2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B2 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_1942_power__le__imp__le__base,axiom,
    ! [A: nat,N2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B2 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_1943_power__le__imp__le__base,axiom,
    ! [A: int,N2: nat,B2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B2 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_1944_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ N2 )
        = zero_z5237406670263579293d_enat ) ) ).

% zero_power
thf(fact_1945_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_nat @ zero_zero_nat @ N2 )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_1946_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ zero_zero_real @ N2 )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_1947_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ zero_zero_complex @ N2 )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_1948_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_int @ zero_zero_int @ N2 )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_1949_power__gt__expt,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).

% power_gt_expt
thf(fact_1950_nat__one__le__power,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).

% nat_one_le_power
thf(fact_1951_power__eq__imp__eq__base,axiom,
    ! [A: real,N2: nat,B2: real] :
      ( ( ( power_power_real @ A @ N2 )
        = ( power_power_real @ B2 @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1952_power__eq__imp__eq__base,axiom,
    ! [A: nat,N2: nat,B2: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = ( power_power_nat @ B2 @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1953_power__eq__imp__eq__base,axiom,
    ! [A: int,N2: nat,B2: int] :
      ( ( ( power_power_int @ A @ N2 )
        = ( power_power_int @ B2 @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1954_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: real,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ( power_power_real @ A @ N2 )
              = ( power_power_real @ B2 @ N2 ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1955_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ( power_power_nat @ A @ N2 )
              = ( power_power_nat @ B2 @ N2 ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1956_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: int,B2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ( power_power_int @ A @ N2 )
              = ( power_power_int @ B2 @ N2 ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1957_zero__power2,axiom,
    ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z5237406670263579293d_enat ) ).

% zero_power2
thf(fact_1958_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_1959_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_1960_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_1961_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_1962_less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% less_exp
thf(fact_1963_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_1964_power2__nat__le__eq__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% power2_nat_le_eq_le
thf(fact_1965_power2__nat__le__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% power2_nat_le_imp_le
thf(fact_1966_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_1967_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_1968_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_1969_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_1970_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_1971_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_1972_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_1973_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_1974_power__strict__mono,axiom,
    ! [A: real,B2: real,N2: nat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B2 @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_1975_power__strict__mono,axiom,
    ! [A: nat,B2: nat,N2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B2 @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_1976_power__strict__mono,axiom,
    ! [A: int,B2: int,N2: nat] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B2 @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_1977_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_1978_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_1979_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_1980_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_1981_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_1982_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_1983_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_1984_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_1985_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_1986_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_1987_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_1988_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_1989_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_1990_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList2: 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 @ TreeList2 ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_1991_add__self__div__2,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M2 ) ).

% add_self_div_2
thf(fact_1992_div2__Suc__Suc,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_1993_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: 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 @ TreeList2 @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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_1994_div__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( divide_divide_nat @ M2 @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1995_div__by__Suc__0,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = M2 ) ).

% div_by_Suc_0
thf(fact_1996_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,M2: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
       => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).

% set_n_deg_not_0
thf(fact_1997_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_1998_Suc__n__div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_1999_div__exp__eq,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2000_div__exp__eq,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2001_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_2002_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_2003_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_2004_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_2005_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_2006_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_2007_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_2008_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_2009_divide__eq__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B2 )
        = one_one_complex )
      = ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_2010_divide__eq__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( divide_divide_real @ A @ B2 )
        = one_one_real )
      = ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_2011_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_2012_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_2013_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_2014_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_2015_one__eq__divide__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_2016_one__eq__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B2 ) )
      = ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_2017_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_2018_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_2019_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_2020_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_2021_divide__eq__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ( divide_divide_real @ B2 @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_eq_1
thf(fact_2022_eq__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B2 @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% eq_divide_eq_1
thf(fact_2023_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_2024_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_2025_power__inject__exp,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M2 )
          = ( power_power_nat @ A @ N2 ) )
        = ( M2 = N2 ) ) ) ).

% power_inject_exp
thf(fact_2026_power__inject__exp,axiom,
    ! [A: real,M2: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M2 )
          = ( power_power_real @ A @ N2 ) )
        = ( M2 = N2 ) ) ) ).

% power_inject_exp
thf(fact_2027_power__inject__exp,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M2 )
          = ( power_power_int @ A @ N2 ) )
        = ( M2 = N2 ) ) ) ).

% power_inject_exp
thf(fact_2028_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_2029_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_2030_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_2031_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_2032_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_2033_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_2034_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_2035_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_2036_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_2037_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_2038_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_2039_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_2040_less__divide__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2041_less__divide__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_real @ B2 @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2042_divide__less__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_real @ B2 @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2043_divide__less__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2044_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_2045_power__strict__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2046_power__strict__increasing__iff,axiom,
    ! [B2: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ ( power_power_real @ B2 @ X ) @ ( power_power_real @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2047_power__strict__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2048_divide__le__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2049_divide__le__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_eq_real @ B2 @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2050_le__divide__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_eq_real @ B2 @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2051_le__divide__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2052_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2053_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2054_one__add__one,axiom,
    ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
    = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2055_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2056_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2057_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ B2 @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N2 ) )
          = ( ord_less_nat @ N2 @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2058_power__strict__decreasing__iff,axiom,
    ! [B2: real,M2: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( ord_less_real @ B2 @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B2 @ M2 ) @ ( power_power_real @ B2 @ N2 ) )
          = ( ord_less_nat @ N2 @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2059_power__strict__decreasing__iff,axiom,
    ! [B2: int,M2: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N2 ) )
          = ( ord_less_nat @ N2 @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2060_power__increasing__iff,axiom,
    ! [B2: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ X ) @ ( power_power_real @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2061_power__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2062_power__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2063_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_2064_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N2 ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2065_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2066_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2067_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2068_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2069_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2070_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2071_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2072_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2073_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2074_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2075_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2076_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2077_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2078_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2079_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2080_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2081_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2082_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_2083_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_2084_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_2085_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_2086_power__decreasing__iff,axiom,
    ! [B2: real,M2: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( ord_less_real @ B2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ M2 ) @ ( power_power_real @ B2 @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2087_power__decreasing__iff,axiom,
    ! [B2: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ B2 @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2088_power__decreasing__iff,axiom,
    ! [B2: int,M2: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2089_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_2090_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_2091_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
      = ( X = one_one_complex ) ) ).

% one_reorient
thf(fact_2092_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_2093_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_2094_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_2095_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_2096_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_2097_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_2098_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_2099_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_2100_zero__neq__one,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_neq_one
thf(fact_2101_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_2102_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).

% less_numeral_extra(4)
thf(fact_2103_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_2104_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_2105_div__add__self1,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_2106_div__add__self1,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_2107_div__add__self2,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_2108_div__add__self2,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_2109_not__one__le__zero,axiom,
    ~ ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).

% not_one_le_zero
thf(fact_2110_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2111_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2112_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2113_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2114_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_2115_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_2116_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_2117_zero__less__one__class_Ozero__le__one,axiom,
    ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% zero_less_one_class.zero_le_one
thf(fact_2118_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_2119_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_2120_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_2121_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2122_not__one__less__zero,axiom,
    ~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).

% not_one_less_zero
thf(fact_2123_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2124_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2125_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2126_zero__less__one,axiom,
    ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% zero_less_one
thf(fact_2127_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2128_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2129_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2130_less__numeral__extra_I1_J,axiom,
    ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% less_numeral_extra(1)
thf(fact_2131_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2132_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2133_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% one_le_numeral
thf(fact_2134_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).

% one_le_numeral
thf(fact_2135_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_2136_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_2137_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_2138_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ).

% not_numeral_less_one
thf(fact_2139_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_2140_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_2141_add__mono1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_2142_add__mono1,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ B2 @ one_on7984719198319812577d_enat ) ) ) ).

% add_mono1
thf(fact_2143_add__mono1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B2 @ one_one_real ) ) ) ).

% add_mono1
thf(fact_2144_add__mono1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).

% add_mono1
thf(fact_2145_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_2146_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_2147_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_2148_right__inverse__eq,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A @ B2 )
          = one_one_complex )
        = ( A = B2 ) ) ) ).

% right_inverse_eq
thf(fact_2149_right__inverse__eq,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B2 )
          = one_one_real )
        = ( A = B2 ) ) ) ).

% right_inverse_eq
thf(fact_2150_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_2151_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_2152_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_2153_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_2154_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_2155_one__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2156_one__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2157_one__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2158_div__eq__dividend__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ( divide_divide_nat @ M2 @ N2 )
          = M2 )
        = ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2159_div__less__dividend,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).

% div_less_dividend
thf(fact_2160_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2161_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2162_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2163_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2164_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2165_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_2166_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_2167_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_2168_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2169_zero__less__two,axiom,
    ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).

% zero_less_two
thf(fact_2170_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2171_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2172_less__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ A @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B2 @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2173_divide__less__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B2 @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ A @ B2 ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_2174_power__le__one,axiom,
    ! [A: real,N2: 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 @ N2 ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_2175_power__le__one,axiom,
    ! [A: nat,N2: 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 @ N2 ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2176_power__le__one,axiom,
    ! [A: int,N2: 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 @ N2 ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2177_less__half__sum,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_2178_gt__half__sum,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B2 ) ) ).

% gt_half_sum
thf(fact_2179_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ N2 )
          = one_on7984719198319812577d_enat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ N2 )
          = zero_z5237406670263579293d_enat ) ) ) ).

% power_0_left
thf(fact_2180_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_2181_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N2 )
          = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N2 )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_2182_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N2 )
          = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N2 )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_2183_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_2184_power__gt1,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2185_power__gt1,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2186_power__gt1,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2187_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2188_power__less__imp__less__exp,axiom,
    ! [A: real,M2: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2189_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2190_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2191_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2192_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2193_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2194_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2195_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2196_nat__induct__non__zero,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_2197_divide__le__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ A @ B2 ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_2198_le__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ A @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2199_power__Suc__le__self,axiom,
    ! [A: real,N2: 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 @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2200_power__Suc__le__self,axiom,
    ! [A: nat,N2: 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 @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2201_power__Suc__le__self,axiom,
    ! [A: int,N2: 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 @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2202_power__Suc__less__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_2203_power__Suc__less__one,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_2204_power__Suc__less__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_2205_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2206_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2207_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2208_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2209_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2210_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2211_power__le__imp__le__exp,axiom,
    ! [A: real,M2: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2212_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2213_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2214_self__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2215_self__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2216_self__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2217_one__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2218_one__less__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2219_one__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2220_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_2221_div__le__mono,axiom,
    ! [M2: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).

% div_le_mono
thf(fact_2222_div__le__dividend,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ).

% div_le_dividend
thf(fact_2223_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N3: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_2224_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N3: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_2225_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( divide_divide_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_2226_Suc__div__le__mono,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_div_le_mono
thf(fact_2227_div__le__mono2,axiom,
    ! [M2: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).

% div_le_mono2
thf(fact_2228_div__greater__zero__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N2 ) )
      = ( ( ord_less_eq_nat @ N2 @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% div_greater_zero_iff
thf(fact_2229_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2230_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2231_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2232_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2233_nat__induct2,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( P @ N3 )
             => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct2
thf(fact_2234_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D4: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D4 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_2235_low__inv,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
        = X ) ) ).

% low_inv
thf(fact_2236_high__inv,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
        = Y ) ) ).

% high_inv
thf(fact_2237_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( ( vEBT_invar_vebt @ A1 @ A22 )
     => ( ( ? [A4: $o,B4: $o] :
              ( A1
              = ( vEBT_Leaf @ A4 @ B4 ) )
         => ( A22
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( A22 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_invar_vebt @ X2 @ N3 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                     => ( ( M3 = N3 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N3 @ M3 ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( A22 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_invar_vebt @ X2 @ N3 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                       => ( ( M3
                            = ( suc @ N3 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N3 @ M3 ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_invar_vebt @ X2 @ N3 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                         => ( ( M3 = N3 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N3 @ M3 ) )
                             => ( ! [I5: nat] :
                                    ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ X6 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I5: nat] :
                                              ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                    = I5 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                & ! [X2: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
                                                        = I5 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X2 )
                                                      & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ( vEBT_invar_vebt @ X2 @ N3 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                           => ( ( M3
                                = ( suc @ N3 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N3 @ M3 ) )
                               => ( ! [I5: nat] :
                                      ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ X6 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I5: nat] :
                                                ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                      = I5 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                  & ! [X2: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
                                                          = I5 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X2 )
                                                        & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_2238_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A12: vEBT_VEBT,A23: nat] :
          ( ( ? [A3: $o,B3: $o] :
                ( A12
                = ( vEBT_Leaf @ A3 @ B3 ) )
            & ( A23
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X4 @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A23
                = ( plus_plus_nat @ N @ N ) )
              & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X4 @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A23
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X4 @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A23
                = ( plus_plus_nat @ N @ N ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X4: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X4 @ N )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X4 @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A23
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X4: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X4 @ N )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_2239_enat__ord__number_I1_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(1)
thf(fact_2240_enat__ord__number_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(2)
thf(fact_2241_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_2242_Leaf__0__not,axiom,
    ! [A: $o,B2: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_2243_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( N2 = one_one_nat )
       => ? [A4: $o,B4: $o] :
            ( T
            = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).

% deg_1_Leafy
thf(fact_2244_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A4: $o,B4: $o] :
          ( T
          = ( vEBT_Leaf @ A4 @ B4 ) ) ) ).

% deg_1_Leaf
thf(fact_2245_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A3: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).

% deg1Leaf
thf(fact_2246_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B2 @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_2247_mult__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_2248_mult__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_2249_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ C )
        = ( times_times_complex @ B2 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_2250_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B2 ) )
      = ( ( C = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_2251_mult__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_2252_mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_2253_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( ( times_times_complex @ C @ A )
        = ( times_times_complex @ C @ B2 ) )
      = ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_2254_mult__eq__0__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ B2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_2255_mult__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ B2 )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_2256_mult__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ B2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_2257_mult__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ B2 )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
thf(fact_2258_mult__eq__0__iff,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ A @ B2 )
        = zero_z5237406670263579293d_enat )
      = ( ( A = zero_z5237406670263579293d_enat )
        | ( B2 = zero_z5237406670263579293d_enat ) ) ) ).

% mult_eq_0_iff
thf(fact_2259_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_2260_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_2261_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_2262_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_2263_mult__zero__right,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% mult_zero_right
thf(fact_2264_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_2265_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_2266_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_2267_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_2268_mult__zero__left,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ zero_z5237406670263579293d_enat @ A )
      = zero_z5237406670263579293d_enat ) ).

% mult_zero_left
thf(fact_2269_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_2270_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_2271_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_2272_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_2273_mult_Oright__neutral,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ one_on7984719198319812577d_enat )
      = A ) ).

% mult.right_neutral
thf(fact_2274_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_2275_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_2276_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_2277_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_2278_mult__1,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ one_on7984719198319812577d_enat @ A )
      = A ) ).

% mult_1
thf(fact_2279_times__divide__eq__left,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ B2 @ C ) @ A )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ B2 @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2280_times__divide__eq__left,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( times_times_real @ ( divide_divide_real @ B2 @ C ) @ A )
      = ( divide_divide_real @ ( times_times_real @ B2 @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2281_divide__divide__eq__left,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2282_divide__divide__eq__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2283_divide__divide__eq__right,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_2284_divide__divide__eq__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_2285_times__divide__eq__right,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2286_times__divide__eq__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2287_mult__cancel2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ( times_times_nat @ M2 @ K )
        = ( times_times_nat @ N2 @ K ) )
      = ( ( M2 = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_2288_mult__cancel1,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M2 )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( M2 = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_2289_mult__0__right,axiom,
    ! [M2: nat] :
      ( ( times_times_nat @ M2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_2290_mult__is__0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( times_times_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_2291_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M2 @ N2 ) )
      = ( ( M2 = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_2292_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( times_times_nat @ M2 @ N2 )
        = one_one_nat )
      = ( ( M2 = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_2293_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_2294_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_2295_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_2296_mult__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( C
        = ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_2297_mult__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( C
        = ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_2298_mult__cancel__right1,axiom,
    ! [C: complex,B2: complex] :
      ( ( C
        = ( times_times_complex @ B2 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_2299_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_2300_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_2301_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_2302_mult__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( C
        = ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_2303_mult__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( C
        = ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_2304_mult__cancel__left1,axiom,
    ! [C: complex,B2: complex] :
      ( ( C
        = ( times_times_complex @ C @ B2 ) )
      = ( ( C = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_2305_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_2306_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_2307_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2308_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B2 ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2309_nonzero__mult__div__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2310_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2311_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2312_nonzero__mult__div__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2313_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2314_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2315_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B2 @ C ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2316_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B2 @ C ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2317_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2318_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2319_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2320_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2321_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2322_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2323_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( ( C = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
          = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2324_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
          = ( divide_divide_real @ A @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2325_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
          = ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2326_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
          = ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2327_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ A @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_2328_div__mult__mult2,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ A @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_2329_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
        = ( divide_divide_nat @ A @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_2330_div__mult__mult1,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( divide_divide_int @ A @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_2331_distrib__right__numeral,axiom,
    ! [A: complex,B2: complex,V: num] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2332_distrib__right__numeral,axiom,
    ! [A: nat,B2: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2333_distrib__right__numeral,axiom,
    ! [A: extended_enat,B2: extended_enat,V: num] :
      ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ ( numera1916890842035813515d_enat @ V ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B2 @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2334_distrib__right__numeral,axiom,
    ! [A: int,B2: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2335_distrib__right__numeral,axiom,
    ! [A: real,B2: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2336_distrib__left__numeral,axiom,
    ! [V: num,B2: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B2 @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2337_distrib__left__numeral,axiom,
    ! [V: num,B2: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2338_distrib__left__numeral,axiom,
    ! [V: num,B2: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B2 @ C ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B2 ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2339_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2340_distrib__left__numeral,axiom,
    ! [V: num,B2: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2341_one__eq__mult__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M2 @ N2 ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_2342_mult__eq__1__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( times_times_nat @ M2 @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_2343_mult__less__cancel2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M2 @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_2344_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_2345_mult__Suc__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( times_times_nat @ M2 @ ( suc @ N2 ) )
      = ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).

% mult_Suc_right
thf(fact_2346_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B2 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_2347_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W2 ) ) @ A )
      = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_2348_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: complex,W2: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W2 ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W2 )
           != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2349_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( ( ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W2 ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( B2
            = ( 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_2350_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B2: complex,W2: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W2 )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
            = B2 ) )
        & ( ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2351_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( ( A
        = ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
            = B2 ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2352_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B2 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_2353_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W2 ) ) @ A )
      = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_2354_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B2 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2355_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B2 ) )
        = ( divide_divide_real @ one_one_real @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2356_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B2 @ ( times_times_complex @ A @ B2 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2357_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ B2 @ ( times_times_real @ A @ B2 ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2358_div__mult__self4,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_2359_div__mult__self4,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_2360_div__mult__self3,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_2361_div__mult__self3,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_2362_div__mult__self2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_2363_div__mult__self2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_2364_div__mult__self1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_2365_div__mult__self1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_2366_one__le__mult__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).

% one_le_mult_iff
thf(fact_2367_mult__le__cancel2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_2368_div__mult__self__is__m,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N2 ) @ N2 )
        = M2 ) ) ).

% div_mult_self_is_m
thf(fact_2369_div__mult__self1__is__m,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M2 ) @ N2 )
        = M2 ) ) ).

% div_mult_self1_is_m
thf(fact_2370_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2371_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2372_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2373_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B2 ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2374_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ A @ B2 ) @ C )
      = ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2375_mult_Oassoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_2376_mult_Oassoc,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_2377_mult_Oassoc,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_2378_mult_Oassoc,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B2 ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_2379_mult_Oassoc,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ A @ B2 ) @ C )
      = ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_2380_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2381_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2382_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2383_mult_Ocommute,axiom,
    ( times_times_complex
    = ( ^ [A3: complex,B3: complex] : ( times_times_complex @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2384_mult_Ocommute,axiom,
    ( times_7803423173614009249d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] : ( times_7803423173614009249d_enat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2385_mult_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_2386_mult_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_2387_mult_Oleft__commute,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( times_times_real @ B2 @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_2388_mult_Oleft__commute,axiom,
    ! [B2: complex,A: complex,C: complex] :
      ( ( times_times_complex @ B2 @ ( times_times_complex @ A @ C ) )
      = ( times_times_complex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_2389_mult_Oleft__commute,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ B2 @ ( times_7803423173614009249d_enat @ A @ C ) )
      = ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_2390_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size(4)
thf(fact_2391_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_2392_mult__right__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_2393_mult__right__cancel,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_2394_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = ( times_times_complex @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_2395_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_2396_mult__left__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_2397_mult__left__cancel,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_2398_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ C @ A )
          = ( times_times_complex @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_2399_no__zero__divisors,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( ( times_times_nat @ A @ B2 )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_2400_no__zero__divisors,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( ( times_times_int @ A @ B2 )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_2401_no__zero__divisors,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( ( times_times_real @ A @ B2 )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_2402_no__zero__divisors,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( ( times_times_complex @ A @ B2 )
         != zero_zero_complex ) ) ) ).

% no_zero_divisors
thf(fact_2403_no__zero__divisors,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( A != zero_z5237406670263579293d_enat )
     => ( ( B2 != zero_z5237406670263579293d_enat )
       => ( ( times_7803423173614009249d_enat @ A @ B2 )
         != zero_z5237406670263579293d_enat ) ) ) ).

% no_zero_divisors
thf(fact_2404_divisors__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ B2 )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_2405_divisors__zero,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ B2 )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_2406_divisors__zero,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ B2 )
        = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_2407_divisors__zero,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ B2 )
        = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divisors_zero
thf(fact_2408_divisors__zero,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ A @ B2 )
        = zero_z5237406670263579293d_enat )
     => ( ( A = zero_z5237406670263579293d_enat )
        | ( B2 = zero_z5237406670263579293d_enat ) ) ) ).

% divisors_zero
thf(fact_2409_mult__not__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ B2 )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_2410_mult__not__zero,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ B2 )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_2411_mult__not__zero,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ B2 )
       != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B2 != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_2412_mult__not__zero,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ B2 )
       != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B2 != zero_zero_complex ) ) ) ).

% mult_not_zero
thf(fact_2413_mult__not__zero,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ A @ B2 )
       != zero_z5237406670263579293d_enat )
     => ( ( A != zero_z5237406670263579293d_enat )
        & ( B2 != zero_z5237406670263579293d_enat ) ) ) ).

% mult_not_zero
thf(fact_2414_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_2415_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_2416_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_2417_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_2418_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ one_on7984719198319812577d_enat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2419_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_2420_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_2421_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_2422_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_2423_mult_Ocomm__neutral,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ one_on7984719198319812577d_enat )
      = A ) ).

% mult.comm_neutral
thf(fact_2424_crossproduct__noteq,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2425_crossproduct__noteq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2426_crossproduct__noteq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2427_crossproduct__noteq,axiom,
    ! [A: complex,B2: complex,C: complex,D: complex] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ D ) )
       != ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2428_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z3: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X @ Z3 ) )
        = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z3 ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2429_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X @ Z3 ) )
        = ( plus_plus_int @ ( times_times_int @ W2 @ Z3 ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2430_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X: real,Z3: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y ) @ ( times_times_real @ X @ Z3 ) )
        = ( plus_plus_real @ ( times_times_real @ W2 @ Z3 ) @ ( times_times_real @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2431_crossproduct__eq,axiom,
    ! [W2: complex,Y: complex,X: complex,Z3: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ W2 @ Y ) @ ( times_times_complex @ X @ Z3 ) )
        = ( plus_plus_complex @ ( times_times_complex @ W2 @ Z3 ) @ ( times_times_complex @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2432_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E2 ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2433_combine__common__factor,axiom,
    ! [A: int,E2: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2434_combine__common__factor,axiom,
    ! [A: real,E2: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2435_combine__common__factor,axiom,
    ! [A: complex,E2: complex,B2: complex,C: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ ( plus_plus_complex @ ( times_times_complex @ B2 @ E2 ) @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2436_combine__common__factor,axiom,
    ! [A: extended_enat,E2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ E2 ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ B2 @ E2 ) @ C ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2437_distrib__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_2438_distrib__right,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_2439_distrib__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_2440_distrib__right,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_2441_distrib__right,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ C )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_2442_distrib__left,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2443_distrib__left,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_2444_distrib__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_2445_distrib__left,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ A @ ( plus_plus_complex @ B2 @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ B2 ) @ ( times_times_complex @ A @ C ) ) ) ).

% distrib_left
thf(fact_2446_distrib__left,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ ( plus_p3455044024723400733d_enat @ B2 @ C ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ B2 ) @ ( times_7803423173614009249d_enat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2447_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2448_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2449_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2450_comm__semiring__class_Odistrib,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2451_comm__semiring__class_Odistrib,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ C )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2452_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2453_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2454_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ A @ ( plus_plus_complex @ B2 @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ B2 ) @ ( times_times_complex @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2455_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2456_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2457_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2458_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z3: complex,W2: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z3 @ W2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z3 ) @ ( times_times_complex @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_2459_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z3: real,W2: real] :
      ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z3 @ W2 ) )
      = ( divide_divide_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_2460_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z3: complex,W2: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z3 @ W2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W2 ) @ ( times_times_complex @ Y @ Z3 ) ) ) ).

% divide_divide_times_eq
thf(fact_2461_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z3: real,W2: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z3 @ W2 ) )
      = ( divide_divide_real @ ( times_times_real @ X @ W2 ) @ ( times_times_real @ Y @ Z3 ) ) ) ).

% divide_divide_times_eq
thf(fact_2462_divide__divide__eq__left_H,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_2463_divide__divide__eq__left_H,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ C @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_2464_Suc__mult__cancel1,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M2 )
        = ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( M2 = N2 ) ) ).

% Suc_mult_cancel1
thf(fact_2465_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2466_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_2467_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_2468_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_2469_le__square,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).

% le_square
thf(fact_2470_le__cube,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).

% le_cube
thf(fact_2471_add__mult__distrib2,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_2472_add__mult__distrib,axiom,
    ! [M2: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% add_mult_distrib
thf(fact_2473_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_2474_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_2475_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_2476_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_2477_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_2478_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_2479_nat__mult__max__left,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M2 @ N2 ) @ Q3 )
      = ( ord_max_nat @ ( times_times_nat @ M2 @ Q3 ) @ ( times_times_nat @ N2 @ Q3 ) ) ) ).

% nat_mult_max_left
thf(fact_2480_nat__mult__max__right,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( times_times_nat @ M2 @ ( ord_max_nat @ N2 @ Q3 ) )
      = ( ord_max_nat @ ( times_times_nat @ M2 @ N2 ) @ ( times_times_nat @ M2 @ Q3 ) ) ) ).

% nat_mult_max_right
thf(fact_2481_mult__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ C @ D )
       => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B2 )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
           => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2482_mult__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2483_mult__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2484_mult__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2485_mult__mono_H,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ C @ D )
       => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
           => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2486_mult__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2487_mult__mono_H,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2488_mult__mono_H,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2489_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2490_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2491_split__mult__pos__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_2492_split__mult__pos__le,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_2493_mult__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2494_mult__left__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2495_mult__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2496_mult__nonpos__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2497_mult__left__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
       => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ C @ A ) @ ( times_7803423173614009249d_enat @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2498_mult__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2499_mult__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2500_mult__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2501_mult__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2502_mult__right__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2503_mult__right__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
       => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2504_mult__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2505_mult__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2506_mult__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2507_mult__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2508_mult__le__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2509_split__mult__neg__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_2510_split__mult__neg__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_2511_split__mult__neg__le,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_2512_mult__nonneg__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2513_mult__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2514_mult__nonneg__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2515_mult__nonneg__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2516_mult__nonneg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2517_mult__nonneg__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2518_mult__nonpos__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2519_mult__nonpos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2520_mult__nonpos__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2521_mult__nonneg__nonpos2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2522_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2523_mult__nonneg__nonpos2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2524_zero__le__mult__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2525_zero__le__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2526_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
       => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ C @ A ) @ ( times_7803423173614009249d_enat @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2527_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2528_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2529_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2530_mult__neg__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2531_mult__neg__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2532_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2533_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2534_mult__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2535_mult__less__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2536_mult__neg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_2537_mult__neg__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_2538_mult__neg__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_2539_mult__pos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_2540_mult__pos__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_2541_mult__pos__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_2542_mult__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2543_mult__pos__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2544_mult__pos__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2545_mult__pos__neg2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_2546_mult__pos__neg2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_2547_mult__pos__neg2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_2548_zero__less__mult__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2549_zero__less__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2550_zero__less__mult__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2551_zero__less__mult__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2552_zero__less__mult__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2553_zero__less__mult__pos2,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2554_zero__less__mult__pos2,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2555_zero__less__mult__pos2,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2556_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_real @ B2 @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2557_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_int @ B2 @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2558_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2559_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_int @ A @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2560_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2561_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2562_mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2563_mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2564_mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2565_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B2 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2566_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B2 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2567_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2568_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2569_mult__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2570_mult__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2571_mult__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2572_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B2 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2573_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B2 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2574_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2575_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2576_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2577_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B2: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B2 @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2578_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B2: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B2 @ ( times_times_int @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2579_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B2: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B2 @ ( times_times_real @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2580_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B2: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
         != ( plus_plus_complex @ B2 @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2581_less__1__mult,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ M2 )
     => ( ( ord_less_nat @ one_one_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2582_less__1__mult,axiom,
    ! [M2: real,N2: real] :
      ( ( ord_less_real @ one_one_real @ M2 )
     => ( ( ord_less_real @ one_one_real @ N2 )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2583_less__1__mult,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_int @ one_one_int @ M2 )
     => ( ( ord_less_int @ one_one_int @ N2 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2584_frac__eq__eq,axiom,
    ! [Y: complex,Z3: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z3 != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X @ Y )
            = ( divide1717551699836669952omplex @ W2 @ Z3 ) )
          = ( ( times_times_complex @ X @ Z3 )
            = ( times_times_complex @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2585_frac__eq__eq,axiom,
    ! [Y: real,Z3: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W2 @ Z3 ) )
          = ( ( times_times_real @ X @ Z3 )
            = ( times_times_real @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2586_divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_2587_divide__eq__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B2 @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B2
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_2588_eq__divide__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( A
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_2589_eq__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B2 ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_2590_divide__eq__imp,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B2
          = ( times_times_complex @ A @ C ) )
       => ( ( divide1717551699836669952omplex @ B2 @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2591_divide__eq__imp,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B2
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B2 @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2592_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = B2 )
       => ( A
          = ( divide1717551699836669952omplex @ B2 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2593_eq__divide__imp,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B2 )
       => ( A
          = ( divide_divide_real @ B2 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2594_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B2 @ C )
          = A )
        = ( B2
          = ( times_times_complex @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2595_nonzero__divide__eq__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B2 @ C )
          = A )
        = ( B2
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2596_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A
          = ( divide1717551699836669952omplex @ B2 @ C ) )
        = ( ( times_times_complex @ A @ C )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2597_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B2 @ C ) )
        = ( ( times_times_real @ A @ C )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2598_power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2599_power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2600_power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2601_power__Suc,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2602_power__Suc,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( power_8040749407984259932d_enat @ A @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ A @ ( power_8040749407984259932d_enat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2603_power__Suc2,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2604_power__Suc2,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2605_power__Suc2,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2606_power__Suc2,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2607_power__Suc2,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( power_8040749407984259932d_enat @ A @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2608_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_mult_less_cancel1
thf(fact_2609_power__add,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_add
thf(fact_2610_power__add,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_add
thf(fact_2611_power__add,axiom,
    ! [A: real,M2: nat,N2: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) ) ) ).

% power_add
thf(fact_2612_power__add,axiom,
    ! [A: complex,M2: nat,N2: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_add
thf(fact_2613_power__add,axiom,
    ! [A: extended_enat,M2: nat,N2: nat] :
      ( ( power_8040749407984259932d_enat @ A @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A @ M2 ) @ ( power_8040749407984259932d_enat @ A @ N2 ) ) ) ).

% power_add
thf(fact_2614_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_2615_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_2616_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_2617_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% Suc_mult_le_cancel1
thf(fact_2618_mult__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ M2 ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).

% mult_Suc
thf(fact_2619_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_2620_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2
        = ( times_times_nat @ M2 @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2621_less__mult__imp__div__less,axiom,
    ! [M2: nat,I: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N2 ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_2622_div__times__less__eq__dividend,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) @ M2 ) ).

% div_times_less_eq_dividend
thf(fact_2623_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) @ M2 ) ).

% times_div_less_eq_dividend
thf(fact_2624_power__odd__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2625_power__odd__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2626_power__odd__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2627_power__odd__eq,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2628_power__odd__eq,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( power_8040749407984259932d_enat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ A @ ( power_8040749407984259932d_enat @ ( power_8040749407984259932d_enat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2629_Suc__double__not__eq__double,axiom,
    ! [M2: nat,N2: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_double_not_eq_double
thf(fact_2630_double__not__eq__Suc__double,axiom,
    ! [M2: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% double_not_eq_Suc_double
thf(fact_2631_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X5 ) )
     => ( ! [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,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ X5 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_2632_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_2633_mult__less__le__imp__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2634_mult__less__le__imp__less,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2635_mult__less__le__imp__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2636_mult__le__less__imp__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2637_mult__le__less__imp__less,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2638_mult__le__less__imp__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2639_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2640_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2641_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2642_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2643_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2644_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2645_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2646_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_eq_int @ A @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2647_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_eq_real @ B2 @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2648_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_eq_int @ B2 @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2649_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2650_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2651_mult__strict__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2652_mult__strict__mono_H,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2653_mult__strict__mono_H,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2654_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2655_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2656_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2657_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2658_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2659_mult__strict__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2660_mult__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2661_mult__strict__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2662_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2663_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2664_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2665_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2666_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2667_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2668_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2669_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_2670_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_2671_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_2672_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_2673_mult__le__one,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ one_on7984719198319812577d_enat )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B2 )
       => ( ( ord_le2932123472753598470d_enat @ B2 @ one_on7984719198319812577d_enat )
         => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ B2 ) @ one_on7984719198319812577d_enat ) ) ) ) ).

% mult_le_one
thf(fact_2674_mult__le__one,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ( ord_less_eq_real @ B2 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_2675_mult__le__one,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_2676_mult__le__one,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_2677_mult__left__le,axiom,
    ! [C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ one_on7984719198319812577d_enat )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
       => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2678_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_2679_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_2680_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_2681_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_2682_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_2683_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_2684_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_2685_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: $o,B2: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X )
          = ( vEBT_Leaf @ $true @ B2 ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X )
              = ( vEBT_Leaf @ A @ B2 ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_2686_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_2687_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_2688_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_2689_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_2690_divide__less__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2691_less__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2692_neg__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% neg_divide_less_eq
thf(fact_2693_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2694_pos__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2695_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% pos_less_divide_eq
thf(fact_2696_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ ( times_times_real @ Z3 @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z3 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2697_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z3: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z3 @ Y ) @ X )
       => ( ord_less_real @ Z3 @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2698_divide__strict__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2699_divide__strict__left__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2700_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_2701_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C )
        = ( numera6690914467698888265omplex @ W2 ) )
      = ( ( ( C != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2702_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B2 @ C )
        = ( numeral_numeral_real @ W2 ) )
      = ( ( ( C != zero_zero_real )
         => ( B2
            = ( 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_2703_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W2 )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2704_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( ( ( numeral_numeral_real @ W2 )
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2705_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B2 ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B2 )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_2706_divide__add__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z3 ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_add_eq_iff
thf(fact_2707_divide__add__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z3 ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_add_eq_iff
thf(fact_2708_add__divide__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z3 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% add_divide_eq_iff
thf(fact_2709_add__divide__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z3 ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% add_divide_eq_iff
thf(fact_2710_add__num__frac,axiom,
    ! [Y: complex,Z3: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z3 @ ( divide1717551699836669952omplex @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2711_add__num__frac,axiom,
    ! [Y: real,Z3: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z3 @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2712_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z3: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z3 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2713_add__frac__num,axiom,
    ! [Y: real,X: real,Z3: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z3 )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2714_add__frac__eq,axiom,
    ! [Y: complex,Z3: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z3 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z3 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z3 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z3 ) ) ) ) ) ).

% add_frac_eq
thf(fact_2715_add__frac__eq,axiom,
    ! [Y: real,Z3: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z3 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) ) ) ) ).

% add_frac_eq
thf(fact_2716_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z3: complex,A: complex,B2: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z3 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2717_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z3: real,A: real,B2: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z3 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2718_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z3: complex,A: complex,B2: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z3 ) @ B2 )
          = B2 ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z3 ) @ B2 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2719_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z3: real,A: real,B2: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z3 ) @ B2 )
          = B2 ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z3 ) @ B2 )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2720_power__gt1__lemma,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2721_power__gt1__lemma,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2722_power__gt1__lemma,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2723_power__less__power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2724_power__less__power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2725_power__less__power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2726_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B2 ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B2 )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_2727_one__less__mult,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).

% one_less_mult
thf(fact_2728_n__less__m__mult__n,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).

% n_less_m_mult_n
thf(fact_2729_n__less__n__mult__m,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M2 ) ) ) ) ).

% n_less_n_mult_m
thf(fact_2730_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_2731_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N2 )
        = ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_2732_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_2733_realpow__pos__nth2,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R3: real] :
          ( ( ord_less_real @ zero_zero_real @ R3 )
          & ( ( power_power_real @ R3 @ ( suc @ N2 ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_2734_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z: real] :
          ( ( ord_less_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2735_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_2736_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_2737_mult__less__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2738_mult__less__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2739_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_2740_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_2741_mult__less__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2742_mult__less__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2743_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_2744_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_2745_mult__le__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2746_mult__le__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2747_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_2748_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_2749_mult__le__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2750_mult__le__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2751_divide__left__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2752_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z3: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ Y ) @ X )
       => ( ord_less_eq_real @ Z3 @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2753_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z3 @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z3 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2754_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% pos_le_divide_eq
thf(fact_2755_pos__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2756_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2757_neg__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% neg_divide_le_eq
thf(fact_2758_divide__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_left_mono
thf(fact_2759_le__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2760_divide__le__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2761_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_2762_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_2763_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B2 @ ( 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 ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2764_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ 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_2765_power__Suc__less,axiom,
    ! [A: nat,N2: 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 @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2766_power__Suc__less,axiom,
    ! [A: real,N2: 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 @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2767_power__Suc__less,axiom,
    ! [A: int,N2: 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 @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2768_mult__2,axiom,
    ! [Z3: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_complex @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_2769_mult__2,axiom,
    ! [Z3: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_nat @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_2770_mult__2,axiom,
    ! [Z3: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_p3455044024723400733d_enat @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_2771_mult__2,axiom,
    ! [Z3: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_int @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_2772_mult__2,axiom,
    ! [Z3: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_real @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_2773_mult__2__right,axiom,
    ! [Z3: complex] :
      ( ( times_times_complex @ Z3 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_2774_mult__2__right,axiom,
    ! [Z3: nat] :
      ( ( times_times_nat @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_2775_mult__2__right,axiom,
    ! [Z3: extended_enat] :
      ( ( times_7803423173614009249d_enat @ Z3 @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_2776_mult__2__right,axiom,
    ! [Z3: int] :
      ( ( times_times_int @ Z3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_2777_mult__2__right,axiom,
    ! [Z3: real] :
      ( ( times_times_real @ Z3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_2778_left__add__twice,axiom,
    ! [A: complex,B2: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_2779_left__add__twice,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_2780_left__add__twice,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ A @ B2 ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_2781_left__add__twice,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_2782_left__add__twice,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_2783_div__nat__eqI,axiom,
    ! [N2: nat,Q3: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q3 ) @ M2 )
     => ( ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ ( suc @ Q3 ) ) )
       => ( ( divide_divide_nat @ M2 @ N2 )
          = Q3 ) ) ) ).

% div_nat_eqI
thf(fact_2784_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N2 @ Q3 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N2 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_2785_split__div,axiom,
    ! [P: nat > $o,M2: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I3: nat,J2: nat] :
              ( ( ord_less_nat @ J2 @ N2 )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J2 ) )
               => ( P @ I3 ) ) ) ) ) ) ).

% split_div
thf(fact_2786_dividend__less__div__times,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).

% dividend_less_div_times
thf(fact_2787_dividend__less__times__div,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_2788_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_2789_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_2790_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_2791_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B2 @ ( 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 ) @ B2 ) )
            & ( ~ ( 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_2792_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ 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_2793_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_2794_split__div_H,axiom,
    ! [P: nat > $o,M2: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q5: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q5 ) @ M2 )
            & ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ ( suc @ Q5 ) ) )
            & ( P @ Q5 ) ) ) ) ).

% split_div'
thf(fact_2795_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X5 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_2796_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X5 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ X5 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X5 ) )
         => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_2797_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,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X5 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X5 ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ X5 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_2798_power2__sum,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_2799_power2__sum,axiom,
    ! [X: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_2800_power2__sum,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_2801_power2__sum,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_2802_power2__sum,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_2803_zero__le__even__power_H,axiom,
    ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2804_zero__le__even__power_H,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2805_nat__bit__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
       => ( ! [N3: nat] :
              ( ( P @ N3 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_bit_induct
thf(fact_2806_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_2807_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N: nat] : ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_2808_realpow__pos__nth,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ( ( power_power_real @ R3 @ N2 )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_2809_realpow__pos__nth__unique,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X5: real] :
            ( ( ord_less_real @ zero_zero_real @ X5 )
            & ( ( power_power_real @ X5 @ N2 )
              = A )
            & ! [Y6: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y6 )
                  & ( ( power_power_real @ Y6 @ N2 )
                    = A ) )
               => ( Y6 = X5 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_2810_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2811_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2812_odd__power__less__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_2813_odd__power__less__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_2814_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_2815_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
          = ( divide_divide_nat @ M2 @ N2 ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_2816_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M2 @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_2817_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_2818_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_2819_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_2820_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_2821_nat__mult__div__cancel1,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
        = ( divide_divide_nat @ M2 @ N2 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_2822_nat__mult__le__cancel1,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_2823_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2824_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2825_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_2826_discrete,axiom,
    ( ord_less_int
    = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_2827_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M2 )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( M2 = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_2828_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_2829_nat__mult__less__cancel1,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_nat @ M2 @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_2830_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M2 )
          = ( times_times_nat @ K @ N2 ) )
        = ( M2 = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_2831_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B2 )
       => ( ( divide_divide_nat @ A @ B2 )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2832_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B2 )
       => ( ( divide_divide_int @ A @ B2 )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2833_div__positive,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_positive
thf(fact_2834_div__positive,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_positive
thf(fact_2835_mult__le__cancel__iff1,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z3 )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ Z3 ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2836_mult__le__cancel__iff1,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2837_mult__le__cancel__iff2,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z3 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ ( times_times_real @ Z3 @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2838_mult__le__cancel__iff2,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z3 @ X ) @ ( times_times_int @ Z3 @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2839_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_2840_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_2841_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X4: nat,N: nat] : ( modulo_modulo_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% low_def
thf(fact_2842_even__succ__div__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_2843_even__succ__div__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_2844_set__decode__Suc,axiom,
    ! [N2: nat,X: nat] :
      ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_2845_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B4 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A4 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ 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 @ TreeList3 ) )
                          & ~ ( ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_2846_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_2847_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_2848_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_2849_length__product,axiom,
    ! [Xs: list_int,Ys: list_VEBT_VEBT] :
      ( ( size_s6639371672096860321T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_2850_length__product,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_2851_length__product,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( size_s7647898544948552527nt_nat @ ( product_int_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_2852_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_2853_length__product,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( size_s2970893825323803983at_int @ ( product_nat_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_2854_length__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_2855_mult__less__iff1,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z3 )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ Z3 ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2856_mult__less__iff1,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2857_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_2858_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_2859_finite__Collect__disjI,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite_finite_real @ ( collect_real @ P ) )
        & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_2860_finite__Collect__disjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_2861_finite__Collect__disjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_2862_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_2863_finite__Collect__disjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_2864_finite__Collect__disjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P ) )
        & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_2865_finite__Collect__disjI,axiom,
    ! [P: extended_enat > $o,Q: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [X4: extended_enat] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
        & ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_2866_finite__Collect__conjI,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( ( finite_finite_real @ ( collect_real @ P ) )
        | ( finite_finite_real @ ( collect_real @ Q ) ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_2867_finite__Collect__conjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_2868_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
          @ ^ [X4: set_nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_2869_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
          @ ^ [X4: nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_2870_finite__Collect__conjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_2871_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
          @ ^ [X4: int] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_2872_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
          @ ^ [X4: extended_enat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_2873_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_2874_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_2875_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_2876_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_2877_dvd__0__left__iff,axiom,
    ! [A: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ zero_z5237406670263579293d_enat @ A )
      = ( A = zero_z5237406670263579293d_enat ) ) ).

% dvd_0_left_iff
thf(fact_2878_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_2879_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_2880_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_2881_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_2882_dvd__0__right,axiom,
    ! [A: extended_enat] : ( dvd_dv3785147216227455552d_enat @ A @ zero_z5237406670263579293d_enat ) ).

% dvd_0_right
thf(fact_2883_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_2884_dvd__add__triv__right__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ A ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_2885_dvd__add__triv__right__iff,axiom,
    ! [A: real,B2: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ A ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_2886_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_2887_dvd__add__triv__left__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_2888_dvd__add__triv__left__iff,axiom,
    ! [A: real,B2: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_2889_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_2890_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_2891_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_2892_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_2893_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_2894_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_2895_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_2896_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_2897_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_2898_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_2899_div__dvd__div,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A ) @ ( divide_divide_nat @ C @ A ) )
          = ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_2900_div__dvd__div,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ A @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A ) @ ( divide_divide_int @ C @ A ) )
          = ( dvd_dvd_int @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_2901_mod__add__self2,axiom,
    ! [A: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_add_self2
thf(fact_2902_mod__add__self2,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_add_self2
thf(fact_2903_mod__add__self1,axiom,
    ! [B2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_add_self1
thf(fact_2904_mod__add__self1,axiom,
    ! [B2: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_add_self1
thf(fact_2905_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M2 @ N2 ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_2906_mod__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( modulo_modulo_nat @ M2 @ N2 )
        = M2 ) ) ).

% mod_less
thf(fact_2907_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B5: set_complex] : ( ord_le211207098394363844omplex @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_2908_finite__Collect__subsets,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite5468666774076196335d_enat
        @ ( collec2260605976452661553d_enat
          @ ^ [B5: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_2909_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_2910_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B5: set_int] : ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_2911_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_2912_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_2913_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_2914_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_2915_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_2916_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_2917_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_2918_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_2919_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_2920_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_2921_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_2922_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
      = ( ( C = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_2923_unit__prod,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_2924_unit__prod,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_2925_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_2926_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_2927_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ ( times_times_real @ C @ A ) ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_2928_dvd__add__times__triv__right__iff,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B2 @ ( times_times_complex @ C @ A ) ) )
      = ( dvd_dvd_complex @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_2929_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B2 ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_2930_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B2 ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_2931_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B2 ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_2932_dvd__add__times__triv__left__iff,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ ( times_times_complex @ C @ A ) @ B2 ) )
      = ( dvd_dvd_complex @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_2933_mod__mult__self1__is__0,axiom,
    ! [B2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A ) @ B2 )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_2934_mod__mult__self1__is__0,axiom,
    ! [B2: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B2 @ A ) @ B2 )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_2935_mod__mult__self2__is__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_2936_mod__mult__self2__is__0,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ B2 )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_2937_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_2938_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_2939_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_2940_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_2941_dvd__div__mult__self,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_2942_dvd__div__mult__self,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_2943_dvd__mult__div__cancel,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ A ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_2944_dvd__mult__div__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ A ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_2945_unit__div,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_2946_unit__div,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_2947_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_2948_unit__div__1__unit,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_2949_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_2950_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_2951_div__add,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_2952_div__add,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_2953_mod__div__trivial,axiom,
    ! [A: nat,B2: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_2954_mod__div__trivial,axiom,
    ! [A: int,B2: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_2955_bits__mod__div__trivial,axiom,
    ! [A: nat,B2: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_2956_bits__mod__div__trivial,axiom,
    ! [A: int,B2: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_2957_mod__mult__self4,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self4
thf(fact_2958_mod__mult__self4,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self4
thf(fact_2959_mod__mult__self3,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self3
thf(fact_2960_mod__mult__self3,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self3
thf(fact_2961_mod__mult__self2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self2
thf(fact_2962_mod__mult__self2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self2
thf(fact_2963_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self1
thf(fact_2964_mod__mult__self1,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self1
thf(fact_2965_dvd__imp__mod__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( modulo_modulo_nat @ B2 @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_2966_dvd__imp__mod__0,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( modulo_modulo_int @ B2 @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_2967_mod__by__Suc__0,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_2968_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_2969_unit__div__mult__self,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_2970_unit__div__mult__self,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_2971_unit__mult__div__div,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B2 @ A ) ) ) ).

% unit_mult_div_div
thf(fact_2972_unit__mult__div__div,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B2 @ A ) ) ) ).

% unit_mult_div_div
thf(fact_2973_even__Suc,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% even_Suc
thf(fact_2974_even__Suc__Suc__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N2 ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_Suc_Suc_iff
thf(fact_2975_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B2 @ N2 ) )
        = ( dvd_dvd_nat @ A @ B2 ) ) ) ).

% pow_divides_pow_iff
thf(fact_2976_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: int,B2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B2 @ N2 ) )
        = ( dvd_dvd_int @ A @ B2 ) ) ) ).

% pow_divides_pow_iff
thf(fact_2977_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M2 ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_mod_mult_self4
thf(fact_2978_Suc__mod__mult__self3,axiom,
    ! [K: nat,N2: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M2 ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_mod_mult_self3
thf(fact_2979_Suc__mod__mult__self2,axiom,
    ! [M2: nat,N2: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_mod_mult_self2
thf(fact_2980_Suc__mod__mult__self1,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_mod_mult_self1
thf(fact_2981_even__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_2982_even__add,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_2983_odd__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_2984_odd__add,axiom,
    ! [A: int,B2: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_2985_odd__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_2986_even__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_2987_mod2__Suc__Suc,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_2988_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N2 ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_2989_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_2990_zero__le__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_2991_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_2992_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_2993_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_2994_power__less__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_2995_power__less__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_2996_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_2997_even__plus__one__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_2998_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_2999_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_3000_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_3001_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_3002_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_3003_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_3004_zero__less__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3005_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3006_even__succ__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3007_even__succ__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3008_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_3009_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_3010_even__succ__div__2,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3011_even__succ__div__2,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3012_even__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_3013_even__power,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_3014_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_3015_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3016_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3017_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3018_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3019_even__succ__mod__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3020_even__succ__mod__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3021_strict__subset__divisors__dvd,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_set_real
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B2 ) ) )
      = ( ( dvd_dvd_real @ A @ B2 )
        & ~ ( dvd_dvd_real @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3022_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B2 ) ) )
      = ( ( dvd_dvd_nat @ A @ B2 )
        & ~ ( dvd_dvd_nat @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3023_strict__subset__divisors__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B2 ) ) )
      = ( ( dvd_dvd_int @ A @ B2 )
        & ~ ( dvd_dvd_int @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3024_less__set__def,axiom,
    ( ord_le2529575680413868914d_enat
    = ( ^ [A5: set_Extended_enat,B5: set_Extended_enat] :
          ( ord_le8499522857272258027enat_o
          @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ A5 )
          @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_3025_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( ord_less_real_o
          @ ^ [X4: real] : ( member_real @ X4 @ A5 )
          @ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_3026_less__set__def,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( ord_less_set_nat_o
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 )
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_3027_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_3028_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ord_less_int_o
          @ ^ [X4: int] : ( member_int @ X4 @ A5 )
          @ ^ [X4: int] : ( member_int @ X4 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_3029_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ B2 )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B2 @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_3030_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = zero_zero_int )
      = ( dvd_dvd_int @ B2 @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_3031_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( modulo_modulo_nat @ B3 @ A3 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3032_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A3: int,B3: int] :
          ( ( modulo_modulo_int @ B3 @ A3 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3033_mod__0__imp__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ B2 )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B2 @ A ) ) ).

% mod_0_imp_dvd
thf(fact_3034_mod__0__imp__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = zero_zero_int )
     => ( dvd_dvd_int @ B2 @ A ) ) ).

% mod_0_imp_dvd
thf(fact_3035_empty__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat
      @ ^ [X4: list_nat] : $false ) ) ).

% empty_def
thf(fact_3036_empty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X4: set_nat] : $false ) ) ).

% empty_def
thf(fact_3037_empty__def,axiom,
    ( bot_bo7653980558646680370d_enat
    = ( collec4429806609662206161d_enat
      @ ^ [X4: extended_enat] : $false ) ) ).

% empty_def
thf(fact_3038_empty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X4: real] : $false ) ) ).

% empty_def
thf(fact_3039_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X4: nat] : $false ) ) ).

% empty_def
thf(fact_3040_empty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X4: int] : $false ) ) ).

% empty_def
thf(fact_3041_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_3042_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_3043_dvd__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_3044_dvd__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ B2 @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_3045_dvd__mod__iff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_3046_dvd__mod__iff,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_3047_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_3048_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_3049_not__finite__existsD,axiom,
    ! [P: real > $o] :
      ( ~ ( finite_finite_real @ ( collect_real @ P ) )
     => ? [X_12: real] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_3050_not__finite__existsD,axiom,
    ! [P: list_nat > $o] :
      ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
     => ? [X_12: list_nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_3051_not__finite__existsD,axiom,
    ! [P: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
     => ? [X_12: set_nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_3052_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_12: nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_3053_not__finite__existsD,axiom,
    ! [P: complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ? [X_12: complex] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_3054_not__finite__existsD,axiom,
    ! [P: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P ) )
     => ? [X_12: int] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_3055_not__finite__existsD,axiom,
    ! [P: extended_enat > $o] :
      ( ~ ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
     => ? [X_12: extended_enat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_3056_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B: set_nat,R: real > nat > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: nat] :
              ( ( member_nat @ X5 @ B )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A3: real] :
                        ( ( member_real @ A3 @ A2 )
                        & ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3057_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B: set_complex,R: real > complex > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite3207457112153483333omplex @ B )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: complex] :
              ( ( member_complex @ X5 @ B )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A3: real] :
                        ( ( member_real @ A3 @ A2 )
                        & ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3058_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B: set_int,R: real > int > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite_finite_int @ B )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: int] :
              ( ( member_int @ X5 @ B )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A3: real] :
                        ( ( member_real @ A3 @ A2 )
                        & ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3059_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B: set_Extended_enat,R: real > extended_enat > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite4001608067531595151d_enat @ B )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A2 )
             => ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ B )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ B )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A3: real] :
                        ( ( member_real @ A3 @ A2 )
                        & ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3060_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: nat] :
              ( ( member_nat @ X5 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3061_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_complex,R: nat > complex > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite3207457112153483333omplex @ B )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: complex] :
              ( ( member_complex @ X5 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3062_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_int,R: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_int @ B )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: int] :
              ( ( member_int @ X5 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3063_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_Extended_enat,R: nat > extended_enat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite4001608067531595151d_enat @ B )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A2 )
             => ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ B )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3064_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B: set_nat,R: complex > nat > $o] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: nat] :
              ( ( member_nat @ X5 @ B )
              & ~ ( finite3207457112153483333omplex
                  @ ( collect_complex
                    @ ^ [A3: complex] :
                        ( ( member_complex @ A3 @ A2 )
                        & ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3065_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B: set_complex,R: complex > complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: complex] :
              ( ( member_complex @ X5 @ B )
              & ~ ( finite3207457112153483333omplex
                  @ ( collect_complex
                    @ ^ [A3: complex] :
                        ( ( member_complex @ A3 @ A2 )
                        & ( R @ A3 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3066_pred__subset__eq,axiom,
    ! [R: set_Extended_enat,S2: set_Extended_enat] :
      ( ( ord_le100613205991271927enat_o
        @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ R )
        @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ S2 ) )
      = ( ord_le7203529160286727270d_enat @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_3067_pred__subset__eq,axiom,
    ! [R: set_real,S2: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X4: real] : ( member_real @ X4 @ R )
        @ ^ [X4: real] : ( member_real @ X4 @ S2 ) )
      = ( ord_less_eq_set_real @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_3068_pred__subset__eq,axiom,
    ! [R: set_set_nat,S2: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ R )
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ S2 ) )
      = ( ord_le6893508408891458716et_nat @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_3069_pred__subset__eq,axiom,
    ! [R: set_nat,S2: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S2 ) )
      = ( ord_less_eq_set_nat @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_3070_pred__subset__eq,axiom,
    ! [R: set_int,S2: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R )
        @ ^ [X4: int] : ( member_int @ X4 @ S2 ) )
      = ( ord_less_eq_set_int @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_3071_less__eq__set__def,axiom,
    ( ord_le7203529160286727270d_enat
    = ( ^ [A5: set_Extended_enat,B5: set_Extended_enat] :
          ( ord_le100613205991271927enat_o
          @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ A5 )
          @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3072_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X4: real] : ( member_real @ X4 @ A5 )
          @ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3073_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 )
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3074_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3075_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X4: int] : ( member_int @ X4 @ A5 )
          @ ^ [X4: int] : ( member_int @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3076_Collect__subset,axiom,
    ! [A2: set_Extended_enat,P: extended_enat > $o] :
      ( ord_le7203529160286727270d_enat
      @ ( collec4429806609662206161d_enat
        @ ^ [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A2 )
            & ( P @ X4 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3077_Collect__subset,axiom,
    ! [A2: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ A2 )
            & ( P @ X4 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3078_Collect__subset,axiom,
    ! [A2: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( member_list_nat @ X4 @ A2 )
            & ( P @ X4 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3079_Collect__subset,axiom,
    ! [A2: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A2 )
            & ( P @ X4 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3080_Collect__subset,axiom,
    ! [A2: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ( P @ X4 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3081_Collect__subset,axiom,
    ! [A2: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ A2 )
            & ( P @ X4 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3082_subset__divisors__dvd,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_set_real
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B2 ) ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_3083_subset__divisors__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B2 ) ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_3084_subset__divisors__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B2 ) ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_3085_Collect__restrict,axiom,
    ! [X8: set_Extended_enat,P: extended_enat > $o] :
      ( ord_le7203529160286727270d_enat
      @ ( collec4429806609662206161d_enat
        @ ^ [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ X8 )
            & ( P @ X4 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_3086_Collect__restrict,axiom,
    ! [X8: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ X8 )
            & ( P @ X4 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_3087_Collect__restrict,axiom,
    ! [X8: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( member_list_nat @ X4 @ X8 )
            & ( P @ X4 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_3088_Collect__restrict,axiom,
    ! [X8: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( member_set_nat @ X4 @ X8 )
            & ( P @ X4 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_3089_Collect__restrict,axiom,
    ! [X8: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ X8 )
            & ( P @ X4 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_3090_Collect__restrict,axiom,
    ! [X8: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ X8 )
            & ( P @ X4 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_3091_prop__restrict,axiom,
    ! [X: extended_enat,Z5: set_Extended_enat,X8: set_Extended_enat,P: extended_enat > $o] :
      ( ( member_Extended_enat @ X @ Z5 )
     => ( ( ord_le7203529160286727270d_enat @ Z5
          @ ( collec4429806609662206161d_enat
            @ ^ [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ X8 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_3092_prop__restrict,axiom,
    ! [X: real,Z5: set_real,X8: set_real,P: real > $o] :
      ( ( member_real @ X @ Z5 )
     => ( ( ord_less_eq_set_real @ Z5
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ X8 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_3093_prop__restrict,axiom,
    ! [X: list_nat,Z5: set_list_nat,X8: set_list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ X @ Z5 )
     => ( ( ord_le6045566169113846134st_nat @ Z5
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ X8 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_3094_prop__restrict,axiom,
    ! [X: set_nat,Z5: set_set_nat,X8: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X @ Z5 )
     => ( ( ord_le6893508408891458716et_nat @ Z5
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ X8 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_3095_prop__restrict,axiom,
    ! [X: nat,Z5: set_nat,X8: set_nat,P: nat > $o] :
      ( ( member_nat @ X @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ X8 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_3096_prop__restrict,axiom,
    ! [X: int,Z5: set_int,X8: set_int,P: int > $o] :
      ( ( member_int @ X @ Z5 )
     => ( ( ord_less_eq_set_int @ Z5
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ X8 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_3097_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_3098_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_3099_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_3100_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_3101_lambda__zero,axiom,
    ( ( ^ [H: extended_enat] : zero_z5237406670263579293d_enat )
    = ( times_7803423173614009249d_enat @ zero_z5237406670263579293d_enat ) ) ).

% lambda_zero
thf(fact_3102_lambda__one,axiom,
    ( ( ^ [X4: nat] : X4 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_3103_lambda__one,axiom,
    ( ( ^ [X4: int] : X4 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_3104_lambda__one,axiom,
    ( ( ^ [X4: real] : X4 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_3105_lambda__one,axiom,
    ( ( ^ [X4: complex] : X4 )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_3106_lambda__one,axiom,
    ( ( ^ [X4: extended_enat] : X4 )
    = ( times_7803423173614009249d_enat @ one_on7984719198319812577d_enat ) ) ).

% lambda_one
thf(fact_3107_max__def__raw,axiom,
    ( ord_max_real
    = ( ^ [A3: real,B3: real] : ( if_real @ ( ord_less_eq_real @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3108_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3109_max__def__raw,axiom,
    ( ord_max_set_int
    = ( ^ [A3: set_int,B3: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3110_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3111_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A3: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3112_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M2 ) ) ) ) ).

% finite_divisors_nat
thf(fact_3113_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B2 )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_3114_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B2 )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_3115_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_3116_bot__empty__eq2,axiom,
    ( bot_bo1565574316222977092_nat_o
    = ( ^ [X4: vEBT_VEBT,Y5: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X4 @ Y5 ) @ bot_bo1642239108664514429BT_nat ) ) ) ).

% bot_empty_eq2
thf(fact_3117_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_3118_bot__empty__eq2,axiom,
    ( bot_bo4898103413517107610_nat_o
    = ( ^ [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y5 ) @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_3119_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P @ K2 )
            & ( ord_less_nat @ K2 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_3120_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_3121_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ N2 @ M2 ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_3122_mod__add__right__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3123_mod__add__right__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3124_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3125_mod__add__left__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3126_mod__add__cong,axiom,
    ! [A: nat,C: nat,A7: nat,B2: nat,B7: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A7 @ C ) )
     => ( ( ( modulo_modulo_nat @ B2 @ C )
          = ( modulo_modulo_nat @ B7 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A7 @ B7 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3127_mod__add__cong,axiom,
    ! [A: int,C: int,A7: int,B2: int,B7: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A7 @ C ) )
     => ( ( ( modulo_modulo_int @ B2 @ C )
          = ( modulo_modulo_int @ B7 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A7 @ B7 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3128_mod__add__eq,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).

% mod_add_eq
thf(fact_3129_mod__add__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% mod_add_eq
thf(fact_3130_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_3131_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_3132_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_3133_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_3134_dvd__0__left,axiom,
    ! [A: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( A = zero_z5237406670263579293d_enat ) ) ).

% dvd_0_left
thf(fact_3135_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A3: real,B3: real] :
          ( ( A3 = zero_zero_real )
         => ( B3 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_3136_dvd__field__iff,axiom,
    ( dvd_dvd_complex
    = ( ^ [A3: complex,B3: complex] :
          ( ( A3 = zero_zero_complex )
         => ( B3 = zero_zero_complex ) ) ) ) ).

% dvd_field_iff
thf(fact_3137_mod__Suc__Suc__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ N2 ) ) ).

% mod_Suc_Suc_eq
thf(fact_3138_mod__Suc__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).

% mod_Suc_eq
thf(fact_3139_dvdE,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ~ ! [K3: nat] :
            ( A
           != ( times_times_nat @ B2 @ K3 ) ) ) ).

% dvdE
thf(fact_3140_dvdE,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ~ ! [K3: int] :
            ( A
           != ( times_times_int @ B2 @ K3 ) ) ) ).

% dvdE
thf(fact_3141_dvdE,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ~ ! [K3: real] :
            ( A
           != ( times_times_real @ B2 @ K3 ) ) ) ).

% dvdE
thf(fact_3142_dvdE,axiom,
    ! [B2: complex,A: complex] :
      ( ( dvd_dvd_complex @ B2 @ A )
     => ~ ! [K3: complex] :
            ( A
           != ( times_times_complex @ B2 @ K3 ) ) ) ).

% dvdE
thf(fact_3143_dvdE,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ B2 @ A )
     => ~ ! [K3: extended_enat] :
            ( A
           != ( times_7803423173614009249d_enat @ B2 @ K3 ) ) ) ).

% dvdE
thf(fact_3144_dvdI,axiom,
    ! [A: nat,B2: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B2 @ K ) )
     => ( dvd_dvd_nat @ B2 @ A ) ) ).

% dvdI
thf(fact_3145_dvdI,axiom,
    ! [A: int,B2: int,K: int] :
      ( ( A
        = ( times_times_int @ B2 @ K ) )
     => ( dvd_dvd_int @ B2 @ A ) ) ).

% dvdI
thf(fact_3146_dvdI,axiom,
    ! [A: real,B2: real,K: real] :
      ( ( A
        = ( times_times_real @ B2 @ K ) )
     => ( dvd_dvd_real @ B2 @ A ) ) ).

% dvdI
thf(fact_3147_dvdI,axiom,
    ! [A: complex,B2: complex,K: complex] :
      ( ( A
        = ( times_times_complex @ B2 @ K ) )
     => ( dvd_dvd_complex @ B2 @ A ) ) ).

% dvdI
thf(fact_3148_dvdI,axiom,
    ! [A: extended_enat,B2: extended_enat,K: extended_enat] :
      ( ( A
        = ( times_7803423173614009249d_enat @ B2 @ K ) )
     => ( dvd_dv3785147216227455552d_enat @ B2 @ A ) ) ).

% dvdI
thf(fact_3149_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B3: nat,A3: nat] :
        ? [K2: nat] :
          ( A3
          = ( times_times_nat @ B3 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_3150_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B3: int,A3: int] :
        ? [K2: int] :
          ( A3
          = ( times_times_int @ B3 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_3151_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B3: real,A3: real] :
        ? [K2: real] :
          ( A3
          = ( times_times_real @ B3 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_3152_dvd__def,axiom,
    ( dvd_dvd_complex
    = ( ^ [B3: complex,A3: complex] :
        ? [K2: complex] :
          ( A3
          = ( times_times_complex @ B3 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_3153_dvd__def,axiom,
    ( dvd_dv3785147216227455552d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
        ? [K2: extended_enat] :
          ( A3
          = ( times_7803423173614009249d_enat @ B3 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_3154_dvd__mult,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_3155_dvd__mult,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_3156_dvd__mult,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_3157_dvd__mult,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( dvd_dvd_complex @ A @ C )
     => ( dvd_dvd_complex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_3158_dvd__mult,axiom,
    ! [A: extended_enat,C: extended_enat,B2: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ A @ C )
     => ( dvd_dv3785147216227455552d_enat @ A @ ( times_7803423173614009249d_enat @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_3159_dvd__mult2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_3160_dvd__mult2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_3161_dvd__mult2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_3162_dvd__mult2,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( dvd_dvd_complex @ A @ B2 )
     => ( dvd_dvd_complex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_3163_dvd__mult2,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ A @ B2 )
     => ( dvd_dv3785147216227455552d_enat @ A @ ( times_7803423173614009249d_enat @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_3164_dvd__mult__left,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3165_dvd__mult__left,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3166_dvd__mult__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
     => ( dvd_dvd_real @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3167_dvd__mult__left,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B2 ) @ C )
     => ( dvd_dvd_complex @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3168_dvd__mult__left,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ ( times_7803423173614009249d_enat @ A @ B2 ) @ C )
     => ( dvd_dv3785147216227455552d_enat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3169_dvd__triv__left,axiom,
    ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_3170_dvd__triv__left,axiom,
    ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_3171_dvd__triv__left,axiom,
    ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_3172_dvd__triv__left,axiom,
    ! [A: complex,B2: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_3173_dvd__triv__left,axiom,
    ! [A: extended_enat,B2: extended_enat] : ( dvd_dv3785147216227455552d_enat @ A @ ( times_7803423173614009249d_enat @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_3174_mult__dvd__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3175_mult__dvd__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3176_mult__dvd__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3177_mult__dvd__mono,axiom,
    ! [A: complex,B2: complex,C: complex,D: complex] :
      ( ( dvd_dvd_complex @ A @ B2 )
     => ( ( dvd_dvd_complex @ C @ D )
       => ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3178_mult__dvd__mono,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ A @ B2 )
     => ( ( dvd_dv3785147216227455552d_enat @ C @ D )
       => ( dvd_dv3785147216227455552d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3179_dvd__mult__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
     => ( dvd_dvd_nat @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_3180_dvd__mult__right,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
     => ( dvd_dvd_int @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_3181_dvd__mult__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
     => ( dvd_dvd_real @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_3182_dvd__mult__right,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B2 ) @ C )
     => ( dvd_dvd_complex @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_3183_dvd__mult__right,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ ( times_7803423173614009249d_enat @ A @ B2 ) @ C )
     => ( dvd_dv3785147216227455552d_enat @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_3184_dvd__triv__right,axiom,
    ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_3185_dvd__triv__right,axiom,
    ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_3186_dvd__triv__right,axiom,
    ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_3187_dvd__triv__right,axiom,
    ! [A: complex,B2: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_3188_dvd__triv__right,axiom,
    ! [A: extended_enat,B2: extended_enat] : ( dvd_dv3785147216227455552d_enat @ A @ ( times_7803423173614009249d_enat @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_3189_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_3190_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_3191_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_3192_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_3193_unit__imp__dvd,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( dvd_dvd_nat @ B2 @ A ) ) ).

% unit_imp_dvd
thf(fact_3194_unit__imp__dvd,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( dvd_dvd_int @ B2 @ A ) ) ).

% unit_imp_dvd
thf(fact_3195_dvd__unit__imp__unit,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3196_dvd__unit__imp__unit,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3197_dvd__add__right__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3198_dvd__add__right__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3199_dvd__add__right__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
        = ( dvd_dvd_real @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3200_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_3201_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_3202_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
        = ( dvd_dvd_real @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_3203_dvd__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_3204_dvd__add,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_3205_dvd__add,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( ( dvd_dvd_real @ A @ C )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_3206_dvd__add,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ A @ B2 )
     => ( ( dvd_dv3785147216227455552d_enat @ A @ C )
       => ( dvd_dv3785147216227455552d_enat @ A @ ( plus_p3455044024723400733d_enat @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_3207_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( ( divide_divide_nat @ A @ C )
            = ( divide_divide_nat @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3208_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( ( divide_divide_int @ A @ C )
            = ( divide_divide_int @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3209_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( dvd_dvd_real @ C @ A )
     => ( ( dvd_dvd_real @ C @ B2 )
       => ( ( ( divide_divide_real @ A @ C )
            = ( divide_divide_real @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3210_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ( divide_divide_nat @ A @ C )
        = ( divide_divide_nat @ B2 @ C ) )
     => ( ( dvd_dvd_nat @ C @ A )
       => ( ( dvd_dvd_nat @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3211_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( divide_divide_int @ A @ C )
        = ( divide_divide_int @ B2 @ C ) )
     => ( ( dvd_dvd_int @ C @ A )
       => ( ( dvd_dvd_int @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3212_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B2 @ C ) )
     => ( ( dvd_dvd_real @ C @ A )
       => ( ( dvd_dvd_real @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3213_div__div__div__same,axiom,
    ! [D: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ D @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B2 @ D ) )
          = ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_3214_div__div__div__same,axiom,
    ! [D: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ D @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B2 @ D ) )
          = ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_3215_mod__less__eq__dividend,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ M2 ) ).

% mod_less_eq_dividend
thf(fact_3216_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_3217_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_3218_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_3219_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_3220_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_3221_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_3222_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_3223_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_3224_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_3225_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_3226_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_3227_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_3228_subset__decode__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M2 ) @ ( nat_set_decode @ N2 ) )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% subset_decode_imp_le
thf(fact_3229_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B2 ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3230_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3231_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3232_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3233_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ B2 )
        = A )
      = ( ( divide_divide_nat @ A @ B2 )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_3234_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = A )
      = ( ( divide_divide_int @ A @ B2 )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_3235_mod__eqE,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B2 @ C ) )
     => ~ ! [D5: int] :
            ( B2
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D5 ) ) ) ) ).

% mod_eqE
thf(fact_3236_div__add1__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3237_div__add1__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3238_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_3239_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_3240_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3241_minf_I10_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Z )
     => ( ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X2 @ S ) ) )
        = ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3242_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3243_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3244_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3245_minf_I9_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Z )
     => ( ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X2 @ S ) )
        = ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3246_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3247_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3248_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3249_pinf_I10_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ X2 )
     => ( ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X2 @ S ) ) )
        = ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3250_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3251_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3252_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3253_pinf_I9_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z: extended_enat] :
    ! [X2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ X2 )
     => ( ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X2 @ S ) )
        = ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3254_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3255_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3256_dvd__div__eq__0__iff,axiom,
    ! [B2: complex,A: complex] :
      ( ( dvd_dvd_complex @ B2 @ A )
     => ( ( ( divide1717551699836669952omplex @ A @ B2 )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3257_dvd__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ( ( ( divide_divide_nat @ A @ B2 )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3258_dvd__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( ( divide_divide_int @ A @ B2 )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3259_dvd__div__eq__0__iff,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ( ( ( divide_divide_real @ A @ B2 )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3260_unit__mult__right__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B2 @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3261_unit__mult__right__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B2 @ A )
          = ( times_times_int @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3262_unit__mult__left__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B2 )
          = ( times_times_nat @ A @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3263_unit__mult__left__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B2 )
          = ( times_times_int @ A @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3264_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3265_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3266_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3267_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3268_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3269_mult__unit__dvd__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3270_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3271_dvd__mult__unit__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3272_is__unit__mult__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_3273_is__unit__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_3274_mod__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) )
          = N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) )
         != N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 )
          = ( suc @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ) ) ).

% mod_Suc
thf(fact_3275_finite__set__decode,axiom,
    ! [N2: nat] : ( finite_finite_nat @ ( nat_set_decode @ N2 ) ) ).

% finite_set_decode
thf(fact_3276_mod__induct,axiom,
    ! [P: nat > $o,N2: nat,P5: nat,M2: nat] :
      ( ( P @ N2 )
     => ( ( ord_less_nat @ N2 @ P5 )
       => ( ( ord_less_nat @ M2 @ P5 )
         => ( ! [N3: nat] :
                ( ( ord_less_nat @ N3 @ P5 )
               => ( ( P @ N3 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P5 ) ) ) )
           => ( P @ M2 ) ) ) ) ) ).

% mod_induct
thf(fact_3277_dvd__div__mult,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B2 @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_3278_dvd__div__mult,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B2 @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_3279_div__mult__swap,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_3280_div__mult__swap,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_3281_div__div__eq__right,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_3282_div__div__eq__right,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B2 @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_3283_dvd__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_3284_dvd__div__mult2__eq,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_3285_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_3286_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_3287_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_3288_div__mult__div__if__dvd,axiom,
    ! [B2: int,A: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_3289_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N2: nat] :
      ( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
     => ( ! [M3: nat,N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ( P @ N3 @ ( modulo_modulo_nat @ M3 @ N3 ) )
             => ( P @ M3 @ N3 ) ) )
       => ( P @ M2 @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_3290_mod__less__divisor,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ N2 ) ) ).

% mod_less_divisor
thf(fact_3291_dvd__div__unit__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B2 ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3292_dvd__div__unit__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B2 ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3293_div__unit__dvd__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3294_div__unit__dvd__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3295_unit__div__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B2 @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_3296_unit__div__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B2 @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_3297_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3298_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3299_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3300_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3301_mod__Suc__le__divisor,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N2 ) ) @ N2 ) ).

% mod_Suc_le_divisor
thf(fact_3302_mod__eq__0D,axiom,
    ! [M2: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ D )
        = zero_zero_nat )
     => ? [Q2: nat] :
          ( M2
          = ( times_times_nat @ D @ Q2 ) ) ) ).

% mod_eq_0D
thf(fact_3303_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N2: nat,M2: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_3304_dvd__power__le,axiom,
    ! [X: real,Y: real,N2: nat,M2: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_3305_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N2: nat,M2: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_3306_dvd__power__le,axiom,
    ! [X: int,Y: int,N2: nat,M2: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_3307_power__le__dvd,axiom,
    ! [A: nat,N2: nat,B2: nat,M2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B2 )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_3308_power__le__dvd,axiom,
    ! [A: real,N2: nat,B2: real,M2: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B2 )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M2 ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_3309_power__le__dvd,axiom,
    ! [A: complex,N2: nat,B2: complex,M2: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B2 )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M2 ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_3310_power__le__dvd,axiom,
    ! [A: int,N2: nat,B2: int,M2: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B2 )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M2 ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_3311_le__imp__power__dvd,axiom,
    ! [M2: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3312_le__imp__power__dvd,axiom,
    ! [M2: nat,N2: nat,A: real] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3313_le__imp__power__dvd,axiom,
    ! [M2: nat,N2: nat,A: complex] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3314_le__imp__power__dvd,axiom,
    ! [M2: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3315_nat__mod__eq__iff,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_3316_dvd__pos__nat,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ M2 @ N2 )
       => ( ord_less_nat @ zero_zero_nat @ M2 ) ) ) ).

% dvd_pos_nat
thf(fact_3317_nat__dvd__not__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N2 )
       => ~ ( dvd_dvd_nat @ N2 @ M2 ) ) ) ).

% nat_dvd_not_less
thf(fact_3318_bezout__add__nat,axiom,
    ! [A: nat,B2: nat] :
    ? [D5: nat,X5: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D5 @ A )
      & ( dvd_dvd_nat @ D5 @ B2 )
      & ( ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D5 ) )
        | ( ( times_times_nat @ B2 @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D5 ) ) ) ) ).

% bezout_add_nat
thf(fact_3319_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B2: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B2 )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y ) @ D ) )
            | ( ( times_times_nat @ B2 @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
         => ? [X5: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B2 ) )
              & ( ( ( times_times_nat @ A @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ Y3 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_3320_parity__cases,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_3321_parity__cases,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_3322_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_3323_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_3324_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs3: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs3 ) @ A2 )
              & ( ( size_s3451745648224563538omplex @ Xs3 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3325_finite__lists__length__eq,axiom,
    ! [A2: set_Extended_enat,N2: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs3: list_Extended_enat] :
              ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs3 ) @ A2 )
              & ( ( size_s3941691890525107288d_enat @ Xs3 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3326_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs3: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs3 ) @ A2 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs3 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3327_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A2 )
              & ( ( size_size_list_nat @ Xs3 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3328_finite__lists__length__eq,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs3: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs3 ) @ A2 )
              & ( ( size_size_list_int @ Xs3 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3329_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_3330_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3331_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3332_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B2 )
       => ( ( modulo_modulo_nat @ A @ B2 )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3333_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B2 )
       => ( ( modulo_modulo_int @ A @ B2 )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3334_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_3335_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_3336_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_3337_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_3338_finite__lists__length__le,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs3: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs3 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs3 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3339_finite__lists__length__le,axiom,
    ! [A2: set_Extended_enat,N2: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs3: list_Extended_enat] :
              ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs3 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3941691890525107288d_enat @ Xs3 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3340_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs3: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs3 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs3 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3341_finite__lists__length__le,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3342_finite__lists__length__le,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs3: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs3 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs3 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3343_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3344_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3345_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3346_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3347_mod__div__decomp,axiom,
    ! [A: nat,B2: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_3348_mod__div__decomp,axiom,
    ! [A: int,B2: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_3349_div__mult__mod__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3350_div__mult__mod__eq,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3351_mod__div__mult__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3352_mod__div__mult__eq,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3353_mod__mult__div__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3354_mod__mult__div__eq,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3355_mult__div__mod__eq,axiom,
    ! [B2: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3356_mult__div__mod__eq,axiom,
    ! [B2: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3357_div__mult1__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3358_div__mult1__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3359_unit__dvdE,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] :
              ( B2
             != ( times_times_nat @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_3360_unit__dvdE,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] :
              ( B2
             != ( times_times_int @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_3361_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X4: nat] : ( P @ ( times_times_nat @ L @ X4 ) ) )
      = ( ? [X4: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3362_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X4: int] : ( P @ ( times_times_int @ L @ X4 ) ) )
      = ( ? [X4: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3363_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X4: real] : ( P @ ( times_times_real @ L @ X4 ) ) )
      = ( ? [X4: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X4 @ zero_zero_real ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3364_unity__coeff__ex,axiom,
    ! [P: complex > $o,L: complex] :
      ( ( ? [X4: complex] : ( P @ ( times_times_complex @ L @ X4 ) ) )
      = ( ? [X4: complex] :
            ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X4 @ zero_zero_complex ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3365_unity__coeff__ex,axiom,
    ! [P: extended_enat > $o,L: extended_enat] :
      ( ( ? [X4: extended_enat] : ( P @ ( times_7803423173614009249d_enat @ L @ X4 ) ) )
      = ( ? [X4: extended_enat] :
            ( ( dvd_dv3785147216227455552d_enat @ L @ ( plus_p3455044024723400733d_enat @ X4 @ zero_z5237406670263579293d_enat ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3366_dvd__div__eq__mult,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B2 )
       => ( ( ( divide_divide_nat @ B2 @ A )
            = C )
          = ( B2
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_3367_dvd__div__eq__mult,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B2 )
       => ( ( ( divide_divide_int @ B2 @ A )
            = C )
          = ( B2
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_3368_div__dvd__iff__mult,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_3369_div__dvd__iff__mult,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_3370_dvd__div__iff__mult,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_3371_dvd__div__iff__mult,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_3372_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B2: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B2 )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B2 @ A )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B2 @ C )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_3373_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B2 )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B2 @ A )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B2 @ C )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_3374_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B2 )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3375_unit__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B2 )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3376_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3377_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3378_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3379_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3380_unit__div__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_3381_unit__div__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_3382_div__mult__unit2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3383_div__mult__unit2,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3384_unit__eq__div2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B2 ) )
        = ( ( times_times_nat @ A @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3385_unit__eq__div2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B2 ) )
        = ( ( times_times_int @ A @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3386_unit__eq__div1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B2 )
          = C )
        = ( A
          = ( times_times_nat @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_3387_unit__eq__div1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B2 )
          = C )
        = ( A
          = ( times_times_int @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_3388_mod__le__divisor,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N2 ) @ N2 ) ) ).

% mod_le_divisor
thf(fact_3389_is__unit__power__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_3390_is__unit__power__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_3391_div__less__mono,axiom,
    ! [A2: nat,B: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A2 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_3392_mod__eq__nat1E,axiom,
    ! [M2: nat,Q3: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q3 )
        = ( modulo_modulo_nat @ N2 @ Q3 ) )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ~ ! [S3: nat] :
              ( M2
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_3393_mod__eq__nat2E,axiom,
    ! [M2: nat,Q3: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q3 )
        = ( modulo_modulo_nat @ N2 @ Q3 ) )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ~ ! [S3: nat] :
              ( N2
             != ( plus_plus_nat @ M2 @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_3394_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q2: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_3395_dvd__imp__le,axiom,
    ! [K: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% dvd_imp_le
thf(fact_3396_div__mod__decomp,axiom,
    ! [A2: nat,N2: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).

% div_mod_decomp
thf(fact_3397_mod__mult2__eq,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( times_times_nat @ N2 @ Q3 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M2 @ N2 ) @ Q3 ) ) @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_3398_dvd__mult__cancel,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M2 @ N2 ) ) ) ).

% dvd_mult_cancel
thf(fact_3399_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
        = ( dvd_dvd_nat @ M2 @ N2 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_3400_bezout__add__strong__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ? [D5: nat,X5: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D5 @ A )
          & ( dvd_dvd_nat @ D5 @ B2 )
          & ( ( times_times_nat @ A @ X5 )
            = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D5 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_3401_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_3402_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_3403_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_3404_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B4 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B4 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3405_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( ( dvd_dvd_int @ B4 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B4 )
                 => ( ( ( divide_divide_int @ one_one_int @ B4 )
                      = A )
                   => ( ( ( times_times_int @ A @ B4 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3406_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B2 ) )
          = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_3407_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B2 ) )
          = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_3408_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_3409_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ A ) )
          = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_3410_odd__even__add,axiom,
    ! [A: nat,B2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_3411_odd__even__add,axiom,
    ! [A: int,B2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_3412_dvd__power__iff,axiom,
    ! [X: nat,M2: nat,N2: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M2 ) @ ( power_power_nat @ X @ N2 ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_3413_dvd__power__iff,axiom,
    ! [X: int,M2: nat,N2: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ N2 ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_3414_dvd__power,axiom,
    ! [N2: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3415_dvd__power,axiom,
    ! [N2: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3416_dvd__power,axiom,
    ! [N2: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3417_dvd__power,axiom,
    ! [N2: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3418_split__mod,axiom,
    ! [P: nat > $o,M2: nat,N2: nat] :
      ( ( P @ ( modulo_modulo_nat @ M2 @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ M2 ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I3: nat,J2: nat] :
              ( ( ord_less_nat @ J2 @ N2 )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J2 ) )
               => ( P @ J2 ) ) ) ) ) ) ).

% split_mod
thf(fact_3419_dvd__mult__cancel1,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N2 ) @ M2 )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_3420_dvd__mult__cancel2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M2 ) @ M2 )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_3421_power__dvd__imp__le,axiom,
    ! [I: nat,M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N2 ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% power_dvd_imp_le
thf(fact_3422_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3423_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3424_product__nth,axiom,
    ! [N2: nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ ( size_s5460976970255530739at_nat @ Ys ) ) )
     => ( ( nth_Pr6744343527793145070at_nat @ ( produc3544356994491977349at_nat @ Xs @ Ys ) @ N2 )
        = ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_s5460976970255530739at_nat @ Ys ) ) ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s5460976970255530739at_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3425_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N2 )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3426_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N2 )
        = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3427_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) @ N2 )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3428_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys ) @ N2 )
        = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3429_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs @ Ys ) @ N2 )
        = ( product_Pair_int_int @ ( nth_int @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3430_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs @ Ys ) @ N2 )
        = ( product_Pair_int_nat @ ( nth_int @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3431_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) @ N2 )
        = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3432_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( product_nat_int @ Xs @ Ys ) @ N2 )
        = ( product_Pair_nat_int @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3433_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys ) @ N2 )
        = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3434_power__mono__odd,axiom,
    ! [N2: nat,A: real,B2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B2 @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_3435_power__mono__odd,axiom,
    ! [N2: nat,A: int,B2: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B2 @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_3436_Suc__times__mod__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N2 ) ) @ M2 )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_3437_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% odd_pos
thf(fact_3438_dvd__power__iff__le,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M2 ) @ ( power_power_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% dvd_power_iff_le
thf(fact_3439_even__unset__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_3440_even__unset__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_3441_even__set__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_3442_even__set__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_3443_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_3444_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo_modulo_int @ A @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_3445_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_3446_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_3447_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_3448_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N2: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_3449_div__exp__mod__exp__eq,axiom,
    ! [A: int,N2: nat,M2: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_3450_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_3451_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_3452_zero__le__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_3453_zero__le__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_3454_zero__le__odd__power,axiom,
    ! [N2: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_3455_zero__le__odd__power,axiom,
    ! [N2: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_3456_zero__le__even__power,axiom,
    ! [N2: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_3457_zero__le__even__power,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_3458_verit__le__mono__div,axiom,
    ! [A2: nat,B: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_3459_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_3460_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_3461_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide_divide_int @ A @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_3462_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: 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 ) ) @ TreeList2 @ 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 @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_3463_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ 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 @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_3464_zero__less__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_3465_zero__less__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_3466_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( 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,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_3467_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A4 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B4 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_3468_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A4 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B4 )
                  & ( 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,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_3469_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_3470_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_3471_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,TreeList3: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ 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 @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) )
         => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_3472_unset__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3473_unset__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3474_set__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3475_set__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3476_power__le__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_3477_power__le__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_3478_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A4 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B4 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ 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 @ TreeList3 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_3479_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,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ 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 @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_3480_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,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ 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 @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_3481_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: 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 ) ) @ TreeList2 @ 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 @ TreeList2 ) )
          & ~ ( ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 ) ) @ TreeList2 @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_3482_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A4 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B4 )
                  & ( 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,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ 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 @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_3483_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( 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,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ 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 @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_3484_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_3485_divmod__digit__1_I1_J,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_3486_finite__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z6: complex] :
              ( ( power_power_complex @ Z6 @ N2 )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_3487_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z6: real] :
              ( ( power_power_real @ Z6 @ N2 )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_3488_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z6: complex] :
              ( ( power_power_complex @ Z6 @ N2 )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_3489_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B4 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A4 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ 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 @ TreeList3 ) )
                              & ~ ( ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_3490_flip__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3491_flip__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3492_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > nat,Y: real > nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_nat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_nat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( times_times_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_nat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3493_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > nat,Y: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_nat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_nat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( times_times_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_nat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3494_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > nat,Y: complex > nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_nat ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_nat ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( times_times_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_nat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3495_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > nat,Y: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_nat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_nat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( times_times_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_nat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3496_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_Extended_enat,X: extended_enat > nat,Y: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_nat ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_nat ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
                & ( ( times_times_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_nat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3497_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > int,Y: real > int] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_int ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_int ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( times_times_int @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_int ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3498_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > int,Y: nat > int] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_int ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_int ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( times_times_int @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_int ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3499_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > int,Y: complex > int] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_int ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_int ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( times_times_int @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_int ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3500_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > int,Y: int > int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_int ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_int ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( times_times_int @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_int ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3501_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_Extended_enat,X: extended_enat > int,Y: extended_enat > int] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_int ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_int ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
                & ( ( times_times_int @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_int ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3502_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > nat,Y: real > nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3503_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > nat,Y: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3504_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > nat,Y: complex > nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_nat ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_nat ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3505_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > nat,Y: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3506_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_Extended_enat,X: extended_enat > nat,Y: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_nat ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_nat ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3507_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3508_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3509_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3510_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3511_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_Extended_enat,X: extended_enat > real,Y: extended_enat > real] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3512_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( 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,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ 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 @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_3513_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A4 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B4 )
                      & ( 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,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ 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 @ TreeList3 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_3514_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ 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 @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_3515_dvd__antisym,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M2 @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M2 )
       => ( M2 = N2 ) ) ) ).

% dvd_antisym
thf(fact_3516_even__flip__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_3517_even__flip__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_3518_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( 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,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_3519_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_3520_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A4 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B4 )
                      & ( 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,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_3521_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,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ 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 @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_3522_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,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ 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 @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_3523_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,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ 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 @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_3524_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_3525_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_3526_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_3527_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_3528_flip__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_3529_flip__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_3530_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_3531_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M2 )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M2 @ N2 )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3532_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M2 )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M2 @ N2 )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3533_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_3534_num_Osize__gen_I2_J,axiom,
    ! [X22: num] :
      ( ( size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_3535_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_3536_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_3537_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_3538_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_3539_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_3540_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_3541_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_3542_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_3543_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_3544_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_3545_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_3546_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_3547_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_3548_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3549_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_3550_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_3551_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_3552_add__diff__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel
thf(fact_3553_add__diff__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel
thf(fact_3554_diff__add__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
      = A ) ).

% diff_add_cancel
thf(fact_3555_diff__add__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
      = A ) ).

% diff_add_cancel
thf(fact_3556_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( minus_minus_nat @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3557_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
      = ( minus_minus_int @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3558_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
      = ( minus_minus_real @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3559_add__diff__cancel__left_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_3560_add__diff__cancel__left_H,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_3561_add__diff__cancel__left_H,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_3562_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3563_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( minus_minus_int @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3564_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( minus_minus_real @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3565_add__diff__cancel__right_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3566_add__diff__cancel__right_H,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3567_add__diff__cancel__right_H,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3568_diff__Suc__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% diff_Suc_Suc
thf(fact_3569_Suc__diff__diff,axiom,
    ! [M2: nat,N2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_3570_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_3571_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_3572_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_3573_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_3574_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_3575_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_3576_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_3577_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_3578_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_3579_of__bool__eq_I1_J,axiom,
    ( ( zero_n1046097342994218471d_enat @ $false )
    = zero_z5237406670263579293d_enat ) ).

% of_bool_eq(1)
thf(fact_3580_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_3581_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_3582_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_3583_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = zero_zero_complex )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_3584_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1046097342994218471d_enat @ P )
        = zero_z5237406670263579293d_enat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_3585_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_3586_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_3587_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le72135733267957522d_enat @ ( zero_n1046097342994218471d_enat @ P ) @ ( zero_n1046097342994218471d_enat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_3588_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_3589_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_3590_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_3591_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = one_one_complex )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_3592_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_3593_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_3594_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_3595_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_3596_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_3597_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_3598_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_3599_signed__take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_3600_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          | Q ) )
      = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_3601_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          | Q ) )
      = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_3602_diff__ge__0__iff__ge,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
      = ( ord_less_eq_real @ B2 @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_3603_diff__ge__0__iff__ge,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
      = ( ord_less_eq_int @ B2 @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_3604_diff__gt__0__iff__gt,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
      = ( ord_less_real @ B2 @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_3605_diff__gt__0__iff__gt,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
      = ( ord_less_int @ B2 @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_3606_le__add__diff__inverse2,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3607_le__add__diff__inverse2,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3608_le__add__diff__inverse2,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3609_le__add__diff__inverse,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3610_le__add__diff__inverse,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3611_le__add__diff__inverse,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3612_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_3613_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_3614_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_3615_diff__add__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_3616_signed__take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_3617_div__diff,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B2 ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).

% div_diff
thf(fact_3618_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_3619_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_3620_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_3621_zero__less__diff,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% zero_less_diff
thf(fact_3622_diff__is__0__eq_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_3623_diff__is__0__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% diff_is_0_eq
thf(fact_3624_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_3625_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_3626_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_3627_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex @ ~ P )
      = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).

% of_bool_not_iff
thf(fact_3628_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real @ ~ P )
      = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).

% of_bool_not_iff
thf(fact_3629_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2684676970156552555ol_int @ ~ P )
      = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).

% of_bool_not_iff
thf(fact_3630_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_3631_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_3632_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_3633_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_3634_Suc__0__mod__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( zero_n2687167440665602831ol_nat
        @ ( N2
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_3635_signed__take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_3636_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_3637_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A2 ) )
        = A2 ) ) ).

% set_encode_inverse
thf(fact_3638_Suc__pred,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% Suc_pred
thf(fact_3639_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_3640_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_3641_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
        = N2 ) ) ).

% Suc_diff_1
thf(fact_3642_even__diff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ).

% even_diff
thf(fact_3643_of__bool__half__eq__0,axiom,
    ! [B2: $o] :
      ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% of_bool_half_eq_0
thf(fact_3644_of__bool__half__eq__0,axiom,
    ! [B2: $o] :
      ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = zero_zero_int ) ).

% of_bool_half_eq_0
thf(fact_3645_odd__Suc__minus__one,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% odd_Suc_minus_one
thf(fact_3646_even__diff__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N2 ) )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ).

% even_diff_nat
thf(fact_3647_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3648_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3649_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_3650_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_3651_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_3652_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_3653_of__bool__eq__iff,axiom,
    ! [P5: $o,Q3: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P5 )
        = ( zero_n2687167440665602831ol_nat @ Q3 ) )
      = ( P5 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_3654_of__bool__eq__iff,axiom,
    ! [P5: $o,Q3: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P5 )
        = ( zero_n2684676970156552555ol_int @ Q3 ) )
      = ( P5 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_3655_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_3656_diff__right__commute,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_3657_diff__right__commute,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_3658_diff__right__commute,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_3659_diff__eq__diff__eq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_3660_diff__eq__diff__eq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_3661_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n3304061248610475627l_real
        @ ( P
          & Q ) )
      = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).

% of_bool_conj
thf(fact_3662_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n1201886186963655149omplex
        @ ( P
          & Q ) )
      = ( times_times_complex @ ( zero_n1201886186963655149omplex @ P ) @ ( zero_n1201886186963655149omplex @ Q ) ) ) ).

% of_bool_conj
thf(fact_3663_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n1046097342994218471d_enat
        @ ( P
          & Q ) )
      = ( times_7803423173614009249d_enat @ ( zero_n1046097342994218471d_enat @ P ) @ ( zero_n1046097342994218471d_enat @ Q ) ) ) ).

% of_bool_conj
thf(fact_3664_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          & Q ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_conj
thf(fact_3665_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          & Q ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_conj
thf(fact_3666_diff__mono,axiom,
    ! [A: real,B2: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_3667_diff__mono,axiom,
    ! [A: int,B2: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_3668_diff__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_3669_diff__left__mono,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_3670_diff__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_3671_diff__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_3672_diff__eq__diff__less__eq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B2 )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_3673_diff__eq__diff__less__eq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B2 )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_3674_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: complex,Z2: complex] : Y4 = Z2 )
    = ( ^ [A3: complex,B3: complex] :
          ( ( minus_minus_complex @ A3 @ B3 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3675_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [A3: int,B3: int] :
          ( ( minus_minus_int @ A3 @ B3 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3676_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: real,Z2: real] : Y4 = Z2 )
    = ( ^ [A3: real,B3: real] :
          ( ( minus_minus_real @ A3 @ B3 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3677_diff__strict__mono,axiom,
    ! [A: real,B2: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_3678_diff__strict__mono,axiom,
    ! [A: int,B2: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_3679_diff__eq__diff__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B2 )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_3680_diff__eq__diff__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B2 )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_3681_diff__strict__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_3682_diff__strict__left__mono,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_3683_diff__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_3684_diff__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_3685_right__diff__distrib_H,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3686_right__diff__distrib_H,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3687_right__diff__distrib_H,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3688_right__diff__distrib_H,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B2 ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3689_left__diff__distrib_H,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3690_left__diff__distrib_H,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3691_left__diff__distrib_H,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B2 @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B2 @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3692_left__diff__distrib_H,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ B2 @ C ) @ A )
      = ( minus_minus_complex @ ( times_times_complex @ B2 @ A ) @ ( times_times_complex @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3693_right__diff__distrib,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3694_right__diff__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3695_right__diff__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B2 ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3696_left__diff__distrib,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_3697_left__diff__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_3698_left__diff__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ C )
      = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_3699_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B2: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A2 @ B2 )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_3700_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B2: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A2 @ B2 )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_3701_diff__eq__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_3702_diff__eq__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_3703_eq__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( A
        = ( minus_minus_int @ C @ B2 ) )
      = ( ( plus_plus_int @ A @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_3704_eq__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( A
        = ( minus_minus_real @ C @ B2 ) )
      = ( ( plus_plus_real @ A @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_3705_add__diff__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_3706_add__diff__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_3707_diff__diff__eq2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_3708_diff__diff__eq2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_3709_diff__add__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_3710_diff__add__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_3711_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3712_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3713_add__implies__diff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3714_add__implies__diff,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( ( plus_plus_int @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3715_add__implies__diff,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ( plus_plus_real @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3716_diff__diff__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3717_diff__diff__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3718_diff__diff__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3719_add__diff__add,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_3720_add__diff__add,axiom,
    ! [A: real,C: real,B2: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_3721_diff__divide__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).

% diff_divide_distrib
thf(fact_3722_dvd__diff,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( dvd_dvd_int @ X @ Z3 )
       => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z3 ) ) ) ) ).

% dvd_diff
thf(fact_3723_dvd__diff,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( dvd_dvd_real @ X @ Z3 )
       => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z3 ) ) ) ) ).

% dvd_diff
thf(fact_3724_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_3725_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_3726_diffs0__imp__equal,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M2 )
          = zero_zero_nat )
       => ( M2 = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_3727_diff__less__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_3728_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_3729_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M2 = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_3730_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_3731_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_3732_diff__le__mono,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_3733_diff__le__self,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).

% diff_le_self
thf(fact_3734_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
          = ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_3735_diff__le__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_3736_Nat_Odiff__cancel,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_3737_diff__cancel2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% diff_cancel2
thf(fact_3738_diff__add__inverse,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse
thf(fact_3739_diff__add__inverse2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse2
thf(fact_3740_diff__mult__distrib2,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% diff_mult_distrib2
thf(fact_3741_diff__mult__distrib,axiom,
    ! [M2: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% diff_mult_distrib
thf(fact_3742_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z3 )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z3 ) @ ( minus_minus_int @ Y @ Z3 ) ) ) ).

% max_diff_distrib_left
thf(fact_3743_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z3 )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z3 ) @ ( minus_minus_real @ Y @ Z3 ) ) ) ).

% max_diff_distrib_left
thf(fact_3744_dvd__diff__nat,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ M2 )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% dvd_diff_nat
thf(fact_3745_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_3746_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_3747_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_3748_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).

% of_bool_less_eq_one
thf(fact_3749_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_3750_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_3751_split__of__bool__asm,axiom,
    ! [P: real > $o,P5: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_real ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3752_split__of__bool__asm,axiom,
    ! [P: complex > $o,P5: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_complex ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3753_split__of__bool__asm,axiom,
    ! [P: extended_enat > $o,P5: $o] :
      ( ( P @ ( zero_n1046097342994218471d_enat @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_on7984719198319812577d_enat ) )
            | ( ~ P5
              & ~ ( P @ zero_z5237406670263579293d_enat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3754_split__of__bool__asm,axiom,
    ! [P: nat > $o,P5: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_nat ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3755_split__of__bool__asm,axiom,
    ! [P: int > $o,P5: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_int ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3756_split__of__bool,axiom,
    ! [P: real > $o,P5: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
      = ( ( P5
         => ( P @ one_one_real ) )
        & ( ~ P5
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_3757_split__of__bool,axiom,
    ! [P: complex > $o,P5: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
      = ( ( P5
         => ( P @ one_one_complex ) )
        & ( ~ P5
         => ( P @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_3758_split__of__bool,axiom,
    ! [P: extended_enat > $o,P5: $o] :
      ( ( P @ ( zero_n1046097342994218471d_enat @ P5 ) )
      = ( ( P5
         => ( P @ one_on7984719198319812577d_enat ) )
        & ( ~ P5
         => ( P @ zero_z5237406670263579293d_enat ) ) ) ) ).

% split_of_bool
thf(fact_3759_split__of__bool,axiom,
    ! [P: nat > $o,P5: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
      = ( ( P5
         => ( P @ one_one_nat ) )
        & ( ~ P5
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_3760_split__of__bool,axiom,
    ! [P: int > $o,P5: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
      = ( ( P5
         => ( P @ one_one_int ) )
        & ( ~ P5
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_3761_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_3762_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_3763_of__bool__def,axiom,
    ( zero_n1046097342994218471d_enat
    = ( ^ [P6: $o] : ( if_Extended_enat @ P6 @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ) ) ).

% of_bool_def
thf(fact_3764_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_3765_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_3766_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_3767_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_3768_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_3769_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_3770_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ( ( minus_minus_nat @ B2 @ A )
            = C )
          = ( B2
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_3771_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_3772_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_3773_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_3774_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_3775_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_3776_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_3777_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_3778_le__add__diff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_3779_diff__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
        = B2 ) ) ).

% diff_add
thf(fact_3780_le__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B2 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_3781_le__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_3782_diff__le__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_3783_diff__le__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_3784_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N2: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
     => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
         => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3785_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3786_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N2: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
         => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3787_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3788_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3789_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3790_less__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B2 ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_3791_less__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B2 ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_3792_diff__less__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_3793_diff__less__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_3794_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ~ ( ord_less_nat @ A @ B2 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3795_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B2: real] :
      ( ~ ( ord_less_real @ A @ B2 )
     => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3796_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B2: int] :
      ( ~ ( ord_less_int @ A @ B2 )
     => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3797_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3798_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3799_eq__add__iff1,axiom,
    ! [A: complex,E2: complex,C: complex,B2: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B2 @ E2 ) @ D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3800_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3801_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3802_eq__add__iff2,axiom,
    ! [A: complex,E2: complex,C: complex,B2: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B2 @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B2 @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3803_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_3804_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_3805_square__diff__square__factored,axiom,
    ! [X: complex,Y: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ ( times_times_complex @ Y @ Y ) )
      = ( times_times_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_complex @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3806_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B2: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B2 ) )
      = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B2 ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_3807_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B2: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B2 ) )
      = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B2 ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_3808_mult__diff__mult,axiom,
    ! [X: complex,Y: complex,A: complex,B2: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ Y ) @ ( times_times_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ X @ ( minus_minus_complex @ Y @ B2 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X @ A ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_3809_dvd__minus__mod,axiom,
    ! [B2: nat,A: nat] : ( dvd_dvd_nat @ B2 @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_3810_dvd__minus__mod,axiom,
    ! [B2: int,A: int] : ( dvd_dvd_int @ B2 @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_3811_diff__less__Suc,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_3812_Suc__diff__Suc,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ N2 @ M2 )
     => ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
        = ( minus_minus_nat @ M2 @ N2 ) ) ) ).

% Suc_diff_Suc
thf(fact_3813_diff__less,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).

% diff_less
thf(fact_3814_Suc__diff__le,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
        = ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% Suc_diff_le
thf(fact_3815_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M2 @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_3816_diff__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_3817_diff__add__0,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_3818_set__encode__eq,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( ( nat_set_encode @ A2 )
            = ( nat_set_encode @ B ) )
          = ( A2 = B ) ) ) ) ).

% set_encode_eq
thf(fact_3819_add__diff__inverse__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M2 @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_3820_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_3821_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_3822_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_3823_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_3824_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_3825_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_3826_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_3827_dvd__minus__self,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N2 @ M2 ) )
      = ( ( ord_less_nat @ N2 @ M2 )
        | ( dvd_dvd_nat @ M2 @ N2 ) ) ) ).

% dvd_minus_self
thf(fact_3828_less__eq__dvd__minus,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( dvd_dvd_nat @ M2 @ N2 )
        = ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_3829_dvd__diffD1,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M2 @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ M2 )
       => ( ( ord_less_eq_nat @ N2 @ M2 )
         => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).

% dvd_diffD1
thf(fact_3830_dvd__diffD,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M2 @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ N2 @ M2 )
         => ( dvd_dvd_nat @ K @ M2 ) ) ) ) ).

% dvd_diffD
thf(fact_3831_mod__geq,axiom,
    ! [M2: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M2 @ N2 )
     => ( ( modulo_modulo_nat @ M2 @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ).

% mod_geq
thf(fact_3832_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M @ N ) @ M @ ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ).

% mod_if
thf(fact_3833_le__mod__geq,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( modulo_modulo_nat @ M2 @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ).

% le_mod_geq
thf(fact_3834_nat__minus__add__max,axiom,
    ! [N2: nat,M2: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M2 ) @ M2 )
      = ( ord_max_nat @ N2 @ M2 ) ) ).

% nat_minus_add_max
thf(fact_3835_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_3836_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_3837_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_3838_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_3839_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3840_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3841_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_3842_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_3843_divide__diff__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z3 ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_diff_eq_iff
thf(fact_3844_divide__diff__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z3 ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_diff_eq_iff
thf(fact_3845_diff__divide__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z3 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% diff_divide_eq_iff
thf(fact_3846_diff__divide__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z3 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% diff_divide_eq_iff
thf(fact_3847_diff__frac__eq,axiom,
    ! [Y: complex,Z3: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z3 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z3 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z3 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z3 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3848_diff__frac__eq,axiom,
    ! [Y: real,Z3: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z3 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3849_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z3: complex,A: complex,B2: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z3 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3850_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z3: real,A: real,B2: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z3 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3851_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_3852_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_3853_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_3854_inf__period_I4_J,axiom,
    ! [D: int,D6: int,T: int] :
      ( ( dvd_dvd_int @ D @ D6 )
     => ! [X2: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3855_inf__period_I4_J,axiom,
    ! [D: real,D6: real,T: real] :
      ( ( dvd_dvd_real @ D @ D6 )
     => ! [X2: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3856_inf__period_I4_J,axiom,
    ! [D: complex,D6: complex,T: complex] :
      ( ( dvd_dvd_complex @ D @ D6 )
     => ! [X2: complex,K4: complex] :
          ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T ) ) )
          = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K4 @ D6 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3857_inf__period_I3_J,axiom,
    ! [D: int,D6: int,T: int] :
      ( ( dvd_dvd_int @ D @ D6 )
     => ! [X2: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3858_inf__period_I3_J,axiom,
    ! [D: real,D6: real,T: real] :
      ( ( dvd_dvd_real @ D @ D6 )
     => ! [X2: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3859_inf__period_I3_J,axiom,
    ! [D: complex,D6: complex,T: complex] :
      ( ( dvd_dvd_complex @ D @ D6 )
     => ! [X2: complex,K4: complex] :
          ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T ) )
          = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K4 @ D6 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3860_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_3861_minus__mult__div__eq__mod,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_3862_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
      = ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_3863_minus__mod__eq__mult__div,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
      = ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_3864_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_3865_minus__mod__eq__div__mult,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_3866_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_3867_minus__div__mult__eq__mod,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_3868_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_3869_diff__Suc__less,axiom,
    ! [N2: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).

% diff_Suc_less
thf(fact_3870_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B2 ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B2 )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D4: nat] :
                ( ( A
                  = ( plus_plus_nat @ B2 @ D4 ) )
                & ~ ( P @ D4 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_3871_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B2 ) )
      = ( ( ( ord_less_nat @ A @ B2 )
         => ( P @ zero_zero_nat ) )
        & ! [D4: nat] :
            ( ( A
              = ( plus_plus_nat @ B2 @ D4 ) )
           => ( P @ D4 ) ) ) ) ).

% nat_diff_split
thf(fact_3872_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_3873_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N2: 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 ) @ N2 ) )
        = ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_3874_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N2: 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 ) @ N2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_3875_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_3876_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_3877_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( M2
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_3878_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_3879_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M2: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( ( modulo_modulo_nat @ M2 @ Q3 )
          = ( modulo_modulo_nat @ N2 @ Q3 ) )
        = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_3880_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( nat_set_encode @ A2 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_3881_exp__div__exp__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N2 @ M2 ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_3882_exp__div__exp__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N2 @ M2 ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_3883_frac__le__eq,axiom,
    ! [Y: real,Z3: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z3 ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_3884_frac__less__eq,axiom,
    ! [Y: real,Z3: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z3 ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_3885_power__diff,axiom,
    ! [A: complex,N2: nat,M2: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M2 @ N2 ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_3886_power__diff,axiom,
    ! [A: nat,N2: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N2 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_3887_power__diff,axiom,
    ! [A: int,N2: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N2 ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_3888_power__diff,axiom,
    ! [A: real,N2: nat,M2: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M2 @ N2 ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_3889_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_3890_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( N2
        = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_3891_div__geq,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ~ ( ord_less_nat @ M2 @ N2 )
       => ( ( divide_divide_nat @ M2 @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).

% div_geq
thf(fact_3892_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M: nat,N: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M @ N )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% div_if
thf(fact_3893_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).

% add_eq_if
thf(fact_3894_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_3895_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_3896_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).

% mult_eq_if
thf(fact_3897_dvd__minus__add,axiom,
    ! [Q3: nat,N2: nat,R2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ Q3 @ N2 )
     => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M2 ) )
       => ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N2 @ Q3 ) )
          = ( dvd_dvd_nat @ M2 @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M2 ) @ Q3 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_3898_mod__nat__eqI,axiom,
    ! [R2: nat,N2: nat,M2: nat] :
      ( ( ord_less_nat @ R2 @ N2 )
     => ( ( ord_less_eq_nat @ R2 @ M2 )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M2 @ R2 ) )
         => ( ( modulo_modulo_nat @ M2 @ N2 )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_3899_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_3900_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M2 ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3901_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M2 ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3902_power__diff__power__eq,axiom,
    ! [A: nat,N2: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3903_power__diff__power__eq,axiom,
    ! [A: int,N2: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3904_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P6: nat,M: nat] : ( if_nat @ ( M = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3905_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P6: int,M: nat] : ( if_int @ ( M = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3906_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P6: real,M: nat] : ( if_real @ ( M = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3907_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P6: complex,M: nat] : ( if_complex @ ( M = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3908_power__eq__if,axiom,
    ( power_8040749407984259932d_enat
    = ( ^ [P6: extended_enat,M: nat] : ( if_Extended_enat @ ( M = zero_zero_nat ) @ one_on7984719198319812577d_enat @ ( times_7803423173614009249d_enat @ P6 @ ( power_8040749407984259932d_enat @ P6 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3909_power__minus__mult,axiom,
    ! [N2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_3910_power__minus__mult,axiom,
    ! [N2: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_3911_power__minus__mult,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_3912_power__minus__mult,axiom,
    ! [N2: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_3913_power__minus__mult,axiom,
    ! [N2: nat,A: extended_enat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_8040749407984259932d_enat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_3914_diff__le__diff__pow,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_3915_le__div__geq,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( divide_divide_nat @ M2 @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).

% le_div_geq
thf(fact_3916_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_3917_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A4: nat] :
          ( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A4 )
         => ( P @ A4 ) )
     => ( ! [A4: nat,B4: $o] :
            ( ( P @ A4 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A4 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3918_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A4: int] :
          ( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A4 )
         => ( P @ A4 ) )
     => ( ! [A4: int,B4: $o] :
            ( ( P @ A4 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A4 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3919_exp__mod__exp,axiom,
    ! [M2: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M2 @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_3920_exp__mod__exp,axiom,
    ! [M2: nat,N2: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M2 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_3921_power2__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3922_power2__diff,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3923_power2__diff,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3924_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_3925_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_3926_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_3927_divmod__digit__1_I2_J,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo_modulo_int @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_3928_even__mask__div__iff_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_3929_even__mask__div__iff_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_3930_even__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_3931_even__mask__div__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_3932_even__mask__div__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_3933_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q3: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ 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 @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3934_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q3: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ 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 @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3935_inrange,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( 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 ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_3936_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_3937_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,A3: int] : ( if_int @ ( N = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_3938_diff__shunt__var,axiom,
    ! [X: set_Extended_enat,Y: set_Extended_enat] :
      ( ( ( minus_925952699566721837d_enat @ X @ Y )
        = bot_bo7653980558646680370d_enat )
      = ( ord_le7203529160286727270d_enat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_3939_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_3940_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_3941_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_3942_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,A3: nat] : ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_3943_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,A3: int] : ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_3944_odd__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( 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 @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_3945_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3946_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_3947_neg__equal__iff__equal,axiom,
    ! [A: int,B2: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_3948_neg__equal__iff__equal,axiom,
    ! [A: real,B2: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_3949_Diff__cancel,axiom,
    ! [A2: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat @ A2 @ A2 )
      = bot_bo7653980558646680370d_enat ) ).

% Diff_cancel
thf(fact_3950_Diff__cancel,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ A2 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_3951_Diff__cancel,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ A2 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_3952_Diff__cancel,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ A2 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_3953_empty__Diff,axiom,
    ! [A2: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat @ bot_bo7653980558646680370d_enat @ A2 )
      = bot_bo7653980558646680370d_enat ) ).

% empty_Diff
thf(fact_3954_empty__Diff,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_3955_empty__Diff,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_3956_empty__Diff,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_3957_Diff__empty,axiom,
    ! [A2: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat @ A2 @ bot_bo7653980558646680370d_enat )
      = A2 ) ).

% Diff_empty
thf(fact_3958_Diff__empty,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% Diff_empty
thf(fact_3959_Diff__empty,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% Diff_empty
thf(fact_3960_Diff__empty,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Diff_empty
thf(fact_3961_finite__Diff2,axiom,
    ! [B: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B ) )
        = ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_Diff2
thf(fact_3962_finite__Diff2,axiom,
    ! [B: set_int,A2: set_int] :
      ( ( finite_finite_int @ B )
     => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B ) )
        = ( finite_finite_int @ A2 ) ) ) ).

% finite_Diff2
thf(fact_3963_finite__Diff2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A2 @ B ) )
        = ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_3964_finite__Diff2,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) )
        = ( finite_finite_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_3965_finite__Diff,axiom,
    ! [A2: set_complex,B: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B ) ) ) ).

% finite_Diff
thf(fact_3966_finite__Diff,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B ) ) ) ).

% finite_Diff
thf(fact_3967_finite__Diff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A2 @ B ) ) ) ).

% finite_Diff
thf(fact_3968_finite__Diff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).

% finite_Diff
thf(fact_3969_Compl__subset__Compl__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B ) )
      = ( ord_less_eq_set_nat @ B @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_3970_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B ) )
      = ( ord_less_eq_set_int @ B @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_3971_Compl__anti__mono,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_3972_Compl__anti__mono,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_3973_zle__diff1__eq,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z3 @ one_one_int ) )
      = ( ord_less_int @ W2 @ Z3 ) ) ).

% zle_diff1_eq
thf(fact_3974_zle__add1__eq__le,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z3 @ one_one_int ) )
      = ( ord_less_eq_int @ W2 @ Z3 ) ) ).

% zle_add1_eq_le
thf(fact_3975_finite__interval__int2,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B2 ) ) ) ) ).

% finite_interval_int2
thf(fact_3976_finite__interval__int3,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B2 ) ) ) ) ).

% finite_interval_int3
thf(fact_3977_finite__interval__int4,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B2 ) ) ) ) ).

% finite_interval_int4
thf(fact_3978_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_3979_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_3980_neg__le__iff__le,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B2 ) ) ).

% neg_le_iff_le
thf(fact_3981_neg__le__iff__le,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B2 ) ) ).

% neg_le_iff_le
thf(fact_3982_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_3983_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_3984_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_3985_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_3986_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_3987_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_3988_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_3989_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_3990_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_3991_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_3992_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_3993_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_3994_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_3995_neg__less__iff__less,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% neg_less_iff_less
thf(fact_3996_neg__less__iff__less,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% neg_less_iff_less
thf(fact_3997_mult__minus__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_3998_mult__minus__left,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_3999_mult__minus__left,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_4000_minus__mult__minus,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( times_times_complex @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_4001_minus__mult__minus,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
      = ( times_times_int @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_4002_minus__mult__minus,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
      = ( times_times_real @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_4003_mult__minus__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_4004_mult__minus__right,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_4005_mult__minus__right,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_4006_minus__add__distrib,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_4007_minus__add__distrib,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_4008_minus__add__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_4009_minus__add__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_4010_add__minus__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_4011_add__minus__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_4012_minus__diff__eq,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) )
      = ( minus_minus_int @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_4013_minus__diff__eq,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) )
      = ( minus_minus_real @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_4014_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4015_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4016_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4017_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4018_Diff__eq__empty__iff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( ( minus_925952699566721837d_enat @ A2 @ B )
        = bot_bo7653980558646680370d_enat )
      = ( ord_le7203529160286727270d_enat @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_4019_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B: set_real] :
      ( ( ( minus_minus_set_real @ A2 @ B )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_4020_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ( minus_minus_set_nat @ A2 @ B )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_4021_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( ( minus_minus_set_int @ A2 @ B )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_4022_atLeastAtMost__iff,axiom,
    ! [I: extended_enat,L: extended_enat,U: extended_enat] :
      ( ( member_Extended_enat @ I @ ( set_or5403411693681687835d_enat @ L @ U ) )
      = ( ( ord_le2932123472753598470d_enat @ L @ I )
        & ( ord_le2932123472753598470d_enat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4023_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_4024_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_4025_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_4026_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_4027_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_4028_Icc__eq__Icc,axiom,
    ! [L: set_nat,H2: set_nat,L3: set_nat,H3: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L @ H2 )
        = ( set_or4548717258645045905et_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_nat @ L @ H2 )
          & ~ ( ord_less_eq_set_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4029_Icc__eq__Icc,axiom,
    ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
      ( ( ( set_or370866239135849197et_int @ L @ H2 )
        = ( set_or370866239135849197et_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_int @ L @ H2 )
          & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4030_Icc__eq__Icc,axiom,
    ! [L: nat,H2: nat,L3: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
        = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H2 )
          & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4031_Icc__eq__Icc,axiom,
    ! [L: int,H2: int,L3: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H2 )
        = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_int @ L @ H2 )
          & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4032_Icc__eq__Icc,axiom,
    ! [L: real,H2: real,L3: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L @ H2 )
        = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_real @ L @ H2 )
          & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4033_take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% take_bit_of_0
thf(fact_4034_take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% take_bit_of_0
thf(fact_4035_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% finite_atLeastAtMost
thf(fact_4036_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_4037_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_4038_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_4039_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_4040_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_4041_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_4042_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_4043_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_4044_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_4045_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_4046_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_4047_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_4048_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_4049_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_4050_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_4051_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_4052_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_4053_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_4054_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_4055_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_4056_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_4057_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_4058_verit__minus__simplify_I3_J,axiom,
    ! [B2: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B2 )
      = ( uminus1482373934393186551omplex @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_4059_verit__minus__simplify_I3_J,axiom,
    ! [B2: int] :
      ( ( minus_minus_int @ zero_zero_int @ B2 )
      = ( uminus_uminus_int @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_4060_verit__minus__simplify_I3_J,axiom,
    ! [B2: real] :
      ( ( minus_minus_real @ zero_zero_real @ B2 )
      = ( uminus_uminus_real @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_4061_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_4062_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_4063_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_4064_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4065_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4066_uminus__add__conv__diff,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( minus_minus_int @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4067_uminus__add__conv__diff,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( minus_minus_real @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4068_diff__minus__eq__add,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( plus_plus_int @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_4069_diff__minus__eq__add,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( plus_plus_real @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_4070_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_4071_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_4072_atLeastatMost__empty__iff,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ( set_or5403411693681687835d_enat @ A @ B2 )
        = bot_bo7653980558646680370d_enat )
      = ( ~ ( ord_le2932123472753598470d_enat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4073_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A @ B2 )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4074_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ( set_or370866239135849197et_int @ A @ B2 )
        = bot_bot_set_set_int )
      = ( ~ ( ord_less_eq_set_int @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4075_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B2 )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4076_atLeastatMost__empty__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B2 )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4077_atLeastatMost__empty__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B2 )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4078_atLeastatMost__empty__iff2,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( bot_bo7653980558646680370d_enat
        = ( set_or5403411693681687835d_enat @ A @ B2 ) )
      = ( ~ ( ord_le2932123472753598470d_enat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4079_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4080_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( bot_bot_set_set_int
        = ( set_or370866239135849197et_int @ A @ B2 ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4081_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B2: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4082_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B2: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4083_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B2: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B2 ) )
      = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4084_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B2 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B2 )
        | ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_eq_set_nat @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4085_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B2: set_int,C: set_int,D: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B2 ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B2 )
        | ( ( ord_less_eq_set_int @ C @ A )
          & ( ord_less_eq_set_int @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4086_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4087_atLeastatMost__subset__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B2 )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4088_atLeastatMost__subset__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B2 )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4089_atLeastatMost__empty,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A )
     => ( ( set_or5403411693681687835d_enat @ A @ B2 )
        = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
thf(fact_4090_atLeastatMost__empty,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B2 )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_4091_atLeastatMost__empty,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B2 )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_4092_atLeastatMost__empty,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B2 )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_4093_infinite__Icc__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B2 ) ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% infinite_Icc_iff
thf(fact_4094_take__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% take_bit_0
thf(fact_4095_take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
      = zero_zero_int ) ).

% take_bit_0
thf(fact_4096_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_4097_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_4098_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_4099_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_4100_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_4101_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_4102_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_4103_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_4104_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_4105_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_4106_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_4107_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_4108_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_4109_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_4110_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_4111_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_4112_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_4113_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_4114_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_4115_take__bit__of__1__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
        = zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4116_take__bit__of__1__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
        = zero_zero_int )
      = ( N2 = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4117_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4118_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4119_neg__numeral__le__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_4120_neg__numeral__le__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_4121_neg__numeral__less__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_num @ N2 @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_4122_neg__numeral__less__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_num @ N2 @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_4123_take__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_Suc_0
thf(fact_4124_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_4125_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_4126_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_4127_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B2 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_4128_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: complex,W2: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_4129_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( ( ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( B2
            = ( 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_4130_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B2: complex,W2: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
            = B2 ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4131_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( ( A
        = ( divide_divide_real @ B2 @ ( 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 ) ) )
            = B2 ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4132_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_4133_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_4134_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W2: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_4135_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: real,W2: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B2 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_4136_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_4137_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_4138_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_4139_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_4140_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_4141_even__take__bit__eq,axiom,
    ! [N2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
      = ( ( N2 = zero_zero_nat )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_4142_even__take__bit__eq,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( ( N2 = zero_zero_nat )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_4143_Suc__div__eq__add3__div__numeral,axiom,
    ! [M2: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_4144_div__Suc__eq__div__add3,axiom,
    ! [M2: nat,N2: nat] :
      ( ( divide_divide_nat @ M2 @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( divide_divide_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_4145_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M2: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_4146_mod__Suc__eq__mod__add3,axiom,
    ! [M2: nat,N2: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( modulo_modulo_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_4147_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_4148_take__bit__Suc__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_4149_take__bit__Suc__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_4150_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_4151_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_4152_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_4153_take__bit__of__exp,axiom,
    ! [M2: nat,N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N2 @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_4154_take__bit__of__exp,axiom,
    ! [M2: nat,N2: nat] :
      ( ( bit_se2923211474154528505it_int @ M2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N2 @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_4155_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4156_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4157_minus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( minus_minus_int @ zero_zero_int @ L )
      = ( uminus_uminus_int @ L ) ) ).

% minus_int_code(2)
thf(fact_4158_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_4159_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W2: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).

% int_distrib(3)
thf(fact_4160_int__distrib_I4_J,axiom,
    ! [W2: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_4161_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I4: int] :
              ( ( ord_less_int @ I4 @ K )
             => ( ( P @ I4 )
               => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_4162_zdvd__mult__cancel,axiom,
    ! [K: int,M2: int,N2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M2 ) @ ( times_times_int @ K @ N2 ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M2 @ N2 ) ) ) ).

% zdvd_mult_cancel
thf(fact_4163_zdvd__imp__le,axiom,
    ! [Z3: int,N2: int] :
      ( ( dvd_dvd_int @ Z3 @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ Z3 @ N2 ) ) ) ).

% zdvd_imp_le
thf(fact_4164_zdvd__not__zless,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ord_less_int @ M2 @ N2 )
       => ~ ( dvd_dvd_int @ N2 @ M2 ) ) ) ).

% zdvd_not_zless
thf(fact_4165_zdvd__antisym__nonneg,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ N2 )
       => ( ( dvd_dvd_int @ M2 @ N2 )
         => ( ( dvd_dvd_int @ N2 @ M2 )
           => ( M2 = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_4166_zdvd__reduce,axiom,
    ! [K: int,N2: int,M2: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N2 @ ( times_times_int @ K @ M2 ) ) )
      = ( dvd_dvd_int @ K @ N2 ) ) ).

% zdvd_reduce
thf(fact_4167_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_4168_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_4169_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_4170_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_4171_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
     => ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% compl_le_swap1
thf(fact_4172_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_4173_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).

% compl_mono
thf(fact_4174_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_4175_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_4176_add1__zle__eq,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z3 )
      = ( ord_less_int @ W2 @ Z3 ) ) ).

% add1_zle_eq
thf(fact_4177_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I4: int] :
              ( ( ord_less_int @ K @ I4 )
             => ( ( P @ I4 )
               => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_4178_le__imp__0__less,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ).

% le_imp_0_less
thf(fact_4179_zless__add1__eq,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z3 @ one_one_int ) )
      = ( ( ord_less_int @ W2 @ Z3 )
        | ( W2 = Z3 ) ) ) ).

% zless_add1_eq
thf(fact_4180_odd__less__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 ) @ zero_zero_int )
      = ( ord_less_int @ Z3 @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_4181_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ( times_times_int @ M2 @ N2 )
          = one_one_int )
        = ( ( M2 = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_4182_zless__imp__add1__zle,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_int @ W2 @ Z3 )
     => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z3 ) ) ).

% zless_imp_add1_zle
thf(fact_4183_int__one__le__iff__zero__less,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z3 )
      = ( ord_less_int @ zero_zero_int @ Z3 ) ) ).

% int_one_le_iff_zero_less
thf(fact_4184_odd__nonzero,axiom,
    ! [Z3: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_4185_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M2: int,N2: int] :
      ( ( ( times_times_int @ M2 @ N2 )
        = one_one_int )
     => ( ( M2 = one_one_int )
        | ( M2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_4186_zmult__eq__1__iff,axiom,
    ! [M2: int,N2: int] :
      ( ( ( times_times_int @ M2 @ N2 )
        = one_one_int )
      = ( ( ( M2 = one_one_int )
          & ( N2 = one_one_int ) )
        | ( ( M2
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N2
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_4187_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_4188_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_4189_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).

% int_distrib(1)
thf(fact_4190_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_4191_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_4192_equation__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( uminus_uminus_int @ B2 ) )
      = ( B2
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_4193_equation__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( uminus_uminus_real @ B2 ) )
      = ( B2
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_4194_minus__equation__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( uminus_uminus_int @ A )
        = B2 )
      = ( ( uminus_uminus_int @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_4195_minus__equation__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( uminus_uminus_real @ A )
        = B2 )
      = ( ( uminus_uminus_real @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_4196_take__bit__add,axiom,
    ! [N2: nat,A: nat,B2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B2 ) ) )
      = ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ A @ B2 ) ) ) ).

% take_bit_add
thf(fact_4197_take__bit__add,axiom,
    ! [N2: nat,A: int,B2: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B2 ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ A @ B2 ) ) ) ).

% take_bit_add
thf(fact_4198_take__bit__tightened,axiom,
    ! [N2: nat,A: nat,B2: nat,M2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
        = ( bit_se2925701944663578781it_nat @ N2 @ B2 ) )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( bit_se2925701944663578781it_nat @ M2 @ A )
          = ( bit_se2925701944663578781it_nat @ M2 @ B2 ) ) ) ) ).

% take_bit_tightened
thf(fact_4199_take__bit__tightened,axiom,
    ! [N2: nat,A: int,B2: int,M2: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
        = ( bit_se2923211474154528505it_int @ N2 @ B2 ) )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( bit_se2923211474154528505it_int @ M2 @ A )
          = ( bit_se2923211474154528505it_int @ M2 @ B2 ) ) ) ) ).

% take_bit_tightened
thf(fact_4200_take__bit__tightened__less__eq__nat,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M2 @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q3 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_4201_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M2 ) @ M2 ) ).

% take_bit_nat_less_eq_self
thf(fact_4202_take__bit__tightened__less__eq__int,axiom,
    ! [M2: nat,N2: nat,K: int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_4203_le__imp__neg__le,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_4204_le__imp__neg__le,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_4205_minus__le__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).

% minus_le_iff
thf(fact_4206_minus__le__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).

% minus_le_iff
thf(fact_4207_le__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_4208_le__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_4209_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_4210_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_4211_minus__less__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).

% minus_less_iff
thf(fact_4212_minus__less__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).

% minus_less_iff
thf(fact_4213_less__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( ord_less_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_4214_less__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( ord_less_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).

% less_minus_iff
thf(fact_4215_square__eq__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_4216_square__eq__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus_uminus_int @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_4217_square__eq__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus_uminus_real @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_4218_minus__mult__commute,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_4219_minus__mult__commute,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_4220_minus__mult__commute,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_4221_is__num__normalize_I8_J,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_4222_is__num__normalize_I8_J,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_4223_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_4224_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_4225_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_4226_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_4227_minus__diff__commute,axiom,
    ! [B2: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_4228_minus__diff__commute,axiom,
    ! [B2: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B2 ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_4229_minus__divide__right,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_divide_right
thf(fact_4230_minus__divide__divide,axiom,
    ! [A: real,B2: real] :
      ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
      = ( divide_divide_real @ A @ B2 ) ) ).

% minus_divide_divide
thf(fact_4231_minus__divide__left,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).

% minus_divide_left
thf(fact_4232_Diff__infinite__finite,axiom,
    ! [T3: set_complex,S2: set_complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S2 )
       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_4233_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_4234_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_4235_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_4236_double__diff,axiom,
    ! [A2: set_nat,B: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( ord_less_eq_set_nat @ B @ C4 )
       => ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_4237_double__diff,axiom,
    ! [A2: set_int,B: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B )
     => ( ( ord_less_eq_set_int @ B @ C4 )
       => ( ( minus_minus_set_int @ B @ ( minus_minus_set_int @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_4238_Diff__subset,axiom,
    ! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ A2 ) ).

% Diff_subset
thf(fact_4239_Diff__subset,axiom,
    ! [A2: set_int,B: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B ) @ A2 ) ).

% Diff_subset
thf(fact_4240_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D6: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C4 )
     => ( ( ord_less_eq_set_nat @ D6 @ B )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ C4 @ D6 ) ) ) ) ).

% Diff_mono
thf(fact_4241_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D6: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C4 )
     => ( ( ord_less_eq_set_int @ D6 @ B )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B ) @ ( minus_minus_set_int @ C4 @ D6 ) ) ) ) ).

% Diff_mono
thf(fact_4242_psubset__imp__ex__mem,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( ord_le2529575680413868914d_enat @ A2 @ B )
     => ? [B4: extended_enat] : ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4243_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B: set_real] :
      ( ( ord_less_set_real @ A2 @ B )
     => ? [B4: real] : ( member_real @ B4 @ ( minus_minus_set_real @ B @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4244_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B )
     => ? [B4: set_nat] : ( member_set_nat @ B4 @ ( minus_2163939370556025621et_nat @ B @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4245_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( ord_less_set_int @ A2 @ B )
     => ? [B4: int] : ( member_int @ B4 @ ( minus_minus_set_int @ B @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4246_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B )
     => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4247_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_4248_take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_4249_take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_4250_infinite__Icc,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B2 ) ) ) ).

% infinite_Icc
thf(fact_4251_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N2: nat,A: int,B2: int] :
      ( ( ( bit_ri631733984087533419it_int @ N2 @ A )
        = ( bit_ri631733984087533419it_int @ N2 @ B2 ) )
      = ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
        = ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ B2 ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_4252_signed__take__bit__take__bit,axiom,
    ! [M2: nat,N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ M2 @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( if_int_int @ ( ord_less_eq_nat @ N2 @ M2 ) @ ( bit_se2923211474154528505it_int @ N2 ) @ ( bit_ri631733984087533419it_int @ M2 ) @ A ) ) ).

% signed_take_bit_take_bit
thf(fact_4253_all__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M: nat] :
            ( ( ord_less_eq_nat @ M @ N2 )
           => ( P @ M ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X4 ) ) ) ) ).

% all_nat_less
thf(fact_4254_ex__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M: nat] :
            ( ( ord_less_eq_nat @ M @ N2 )
            & ( P @ M ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X4 ) ) ) ) ).

% ex_nat_less
thf(fact_4255_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_4256_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_4257_neg__numeral__le__numeral,axiom,
    ! [M2: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_4258_neg__numeral__le__numeral,axiom,
    ! [M2: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_4259_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_4260_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_4261_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_4262_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_4263_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_4264_neg__numeral__less__numeral,axiom,
    ! [M2: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_4265_neg__numeral__less__numeral,axiom,
    ! [M2: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_4266_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_4267_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_4268_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_4269_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_4270_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_4271_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_4272_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_4273_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B2 )
      = ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4274_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( ( uminus_uminus_int @ A )
        = B2 )
      = ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4275_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( ( uminus_uminus_real @ A )
        = B2 )
      = ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4276_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4277_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( uminus_uminus_int @ B2 ) )
      = ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4278_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( uminus_uminus_real @ B2 ) )
      = ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4279_add_Oinverse__unique,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_4280_add_Oinverse__unique,axiom,
    ! [A: int,B2: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_4281_add_Oinverse__unique,axiom,
    ! [A: real,B2: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_4282_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_4283_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_4284_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_4285_add__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex )
      = ( B2
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4286_add__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int )
      = ( B2
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4287_add__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real )
      = ( B2
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4288_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_4289_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_4290_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_4291_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_4292_nonzero__minus__divide__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_4293_nonzero__minus__divide__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_4294_nonzero__minus__divide__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_4295_nonzero__minus__divide__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_4296_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_4297_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_4298_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_4299_group__cancel_Osub2,axiom,
    ! [B: int,K: int,B2: int,A: int] :
      ( ( B
        = ( plus_plus_int @ K @ B2 ) )
     => ( ( minus_minus_int @ A @ B )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_4300_group__cancel_Osub2,axiom,
    ! [B: real,K: real,B2: real,A: real] :
      ( ( B
        = ( plus_plus_real @ K @ B2 ) )
     => ( ( minus_minus_real @ A @ B )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_4301_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4302_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4303_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4304_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4305_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M2: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
          = ( bit_se4205575877204974255it_nat @ M2 @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_4306_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M2: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
          = ( bit_se4203085406695923979it_int @ M2 @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_4307_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M2: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
          = ( bit_se7882103937844011126it_nat @ M2 @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_4308_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M2: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
          = ( bit_se7879613467334960850it_int @ M2 @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_4309_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M2: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
          = ( bit_se2161824704523386999it_nat @ M2 @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_4310_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M2: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
          = ( bit_se2159334234014336723it_int @ M2 @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_4311_dvd__div__neg,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_4312_dvd__div__neg,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_4313_dvd__neg__div,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_4314_dvd__neg__div,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_4315_subset__Compl__self__eq,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ ( uminus417252749190364093d_enat @ A2 ) )
      = ( A2 = bot_bo7653980558646680370d_enat ) ) ).

% subset_Compl_self_eq
thf(fact_4316_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_4317_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_4318_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_4319_atLeastatMost__psubset__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le2529575680413868914d_enat @ ( set_or5403411693681687835d_enat @ A @ B2 ) @ ( set_or5403411693681687835d_enat @ C @ D ) )
      = ( ( ~ ( ord_le2932123472753598470d_enat @ A @ B2 )
          | ( ( ord_le2932123472753598470d_enat @ C @ A )
            & ( ord_le2932123472753598470d_enat @ B2 @ D )
            & ( ( ord_le72135733267957522d_enat @ C @ A )
              | ( ord_le72135733267957522d_enat @ B2 @ D ) ) ) )
        & ( ord_le2932123472753598470d_enat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4320_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B2 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B2 )
          | ( ( ord_less_eq_set_nat @ C @ A )
            & ( ord_less_eq_set_nat @ B2 @ D )
            & ( ( ord_less_set_nat @ C @ A )
              | ( ord_less_set_nat @ B2 @ D ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4321_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B2: set_int,C: set_int,D: set_int] :
      ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B2 ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_int @ A @ B2 )
          | ( ( ord_less_eq_set_int @ C @ A )
            & ( ord_less_eq_set_int @ B2 @ D )
            & ( ( ord_less_set_int @ C @ A )
              | ( ord_less_set_int @ B2 @ D ) ) ) )
        & ( ord_less_eq_set_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4322_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B2 )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B2 @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B2 @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4323_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B2 )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B2 @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B2 @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4324_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B2 )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B2 @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B2 @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4325_take__bit__signed__take__bit,axiom,
    ! [M2: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( bit_se2923211474154528505it_int @ M2 @ ( bit_ri631733984087533419it_int @ N2 @ A ) )
        = ( bit_se2923211474154528505it_int @ M2 @ A ) ) ) ).

% take_bit_signed_take_bit
thf(fact_4326_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_4327_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_4328_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_4329_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_4330_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_4331_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_4332_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_4333_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_4334_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_4335_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_4336_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_4337_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_4338_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).

% numeral_Bit1
thf(fact_4339_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_4340_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_Bit1
thf(fact_4341_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_4342_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_4343_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_4344_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_4345_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_4346_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_4347_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_4348_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_4349_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_4350_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_4351_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_4352_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_4353_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_4354_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_4355_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_4356_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_4357_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_4358_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_4359_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_4360_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_4361_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_4362_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_4363_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_4364_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_4365_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_4366_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_4367_eq__minus__divide__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = ( uminus1482373934393186551omplex @ B2 ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4368_eq__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A
        = ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = ( uminus_uminus_real @ B2 ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4369_minus__divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B2 )
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4370_minus__divide__eq__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B2 )
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4371_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: complex,A: complex,C: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A )
          = ( times_times_complex @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4372_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( B2 != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
          = C )
        = ( ( uminus_uminus_real @ A )
          = ( times_times_real @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4373_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
        = ( ( times_times_complex @ C @ B2 )
          = ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4374_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) )
        = ( ( times_times_real @ C @ B2 )
          = ( uminus_uminus_real @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4375_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B2 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B2 != zero_zero_complex )
        & ( A
          = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_4376_divide__eq__minus__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( divide_divide_real @ A @ B2 )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B2 != zero_zero_real )
        & ( A
          = ( uminus_uminus_real @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_4377_eval__nat__numeral_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_4378_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_4379_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_4380_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_4381_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_4382_take__bit__Suc__minus__1__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_4383_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).

% numeral_code(3)
thf(fact_4384_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_4385_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_code(3)
thf(fact_4386_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_4387_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_4388_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N6 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_4389_pos__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4390_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4391_neg__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4392_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_4393_minus__divide__less__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_4394_less__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_4395_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_4396_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B2 @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B2
            = ( 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_4397_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4398_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4399_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_4400_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_4401_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z3: complex,A: complex,B2: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z3 ) ) @ B2 )
          = B2 ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z3 ) ) @ B2 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4402_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z3: real,A: real,B2: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z3 ) ) @ B2 )
          = B2 ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z3 ) ) @ B2 )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4403_minus__divide__add__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z3 ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4404_minus__divide__add__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z3 ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4405_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z3: complex,A: complex,B2: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z3 ) ) @ B2 )
          = ( uminus1482373934393186551omplex @ B2 ) ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z3 ) ) @ B2 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4406_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z3: real,A: real,B2: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z3 ) ) @ B2 )
          = ( uminus_uminus_real @ B2 ) ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z3 ) ) @ B2 )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4407_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z3: complex,A: complex,B2: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z3 ) @ B2 )
          = ( uminus1482373934393186551omplex @ B2 ) ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z3 ) @ B2 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4408_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z3: real,A: real,B2: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z3 ) @ B2 )
          = ( uminus_uminus_real @ B2 ) ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z3 ) @ B2 )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4409_minus__divide__diff__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z3 ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4410_minus__divide__diff__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z3 ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4411_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_4412_Suc3__eq__add__3,axiom,
    ! [N2: nat] :
      ( ( suc @ ( suc @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).

% Suc3_eq_add_3
thf(fact_4413_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4414_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4415_take__bit__nat__eq__self,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ M2 )
        = M2 ) ) ).

% take_bit_nat_eq_self
thf(fact_4416_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_nat_less_exp
thf(fact_4417_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ M2 )
        = M2 )
      = ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_4418_num_Osize_I6_J,axiom,
    ! [X32: num] :
      ( ( size_size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_4419_num_Osize__gen_I3_J,axiom,
    ! [X32: num] :
      ( ( size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_4420_le__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4421_minus__divide__le__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4422_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4423_neg__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4424_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4425_pos__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4426_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B2 @ ( 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 ) @ B2 ) )
            & ( ~ ( 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_4427_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ 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_4428_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_4429_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_4430_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_4431_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_4432_Suc__div__eq__add3__div,axiom,
    ! [M2: nat,N2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N2 )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N2 ) ) ).

% Suc_div_eq_add3_div
thf(fact_4433_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4434_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4435_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4436_Suc__mod__eq__add3__mod,axiom,
    ! [M2: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N2 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_4437_take__bit__eq__0__iff,axiom,
    ! [N2: nat,A: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_4438_take__bit__eq__0__iff,axiom,
    ! [N2: nat,A: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
        = zero_zero_int )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_4439_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M2 ) @ M2 )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M2 ) ) ).

% take_bit_nat_less_self_iff
thf(fact_4440_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B2 @ ( 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 ) @ B2 ) )
            & ( ~ ( 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_4441_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ 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_4442_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_4443_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_4444_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_4445_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_4446_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_4447_take__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4448_take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4449_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
            = zero_zero_nat ) )
        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
            = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4450_stable__imp__take__bit__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N2 @ A )
            = zero_zero_int ) )
        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N2 @ A )
            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4451_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_4452_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_num @ M2 @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4453_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_num @ M2 @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4454_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_num @ M2 @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4455_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_num @ M2 @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4456_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4457_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4458_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_4459_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_4460_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_4461_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_4462_set__encode__insert,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ N2 @ A2 )
       => ( ( nat_set_encode @ ( insert_nat @ N2 @ A2 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).

% set_encode_insert
thf(fact_4463_insert__absorb2,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
      = ( insert_nat @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_4464_insert__absorb2,axiom,
    ! [X: int,A2: set_int] :
      ( ( insert_int @ X @ ( insert_int @ X @ A2 ) )
      = ( insert_int @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_4465_insert__absorb2,axiom,
    ! [X: real,A2: set_real] :
      ( ( insert_real @ X @ ( insert_real @ X @ A2 ) )
      = ( insert_real @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_4466_insert__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,A2: set_Extended_enat] :
      ( ( member_Extended_enat @ A @ ( insert_Extended_enat @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member_Extended_enat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4467_insert__iff,axiom,
    ! [A: real,B2: real,A2: set_real] :
      ( ( member_real @ A @ ( insert_real @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member_real @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4468_insert__iff,axiom,
    ! [A: set_nat,B2: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member_set_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4469_insert__iff,axiom,
    ! [A: nat,B2: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4470_insert__iff,axiom,
    ! [A: int,B2: int,A2: set_int] :
      ( ( member_int @ A @ ( insert_int @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member_int @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4471_insertCI,axiom,
    ! [A: extended_enat,B: set_Extended_enat,B2: extended_enat] :
      ( ( ~ ( member_Extended_enat @ A @ B )
       => ( A = B2 ) )
     => ( member_Extended_enat @ A @ ( insert_Extended_enat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_4472_insertCI,axiom,
    ! [A: real,B: set_real,B2: real] :
      ( ( ~ ( member_real @ A @ B )
       => ( A = B2 ) )
     => ( member_real @ A @ ( insert_real @ B2 @ B ) ) ) ).

% insertCI
thf(fact_4473_insertCI,axiom,
    ! [A: set_nat,B: set_set_nat,B2: set_nat] :
      ( ( ~ ( member_set_nat @ A @ B )
       => ( A = B2 ) )
     => ( member_set_nat @ A @ ( insert_set_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_4474_insertCI,axiom,
    ! [A: nat,B: set_nat,B2: nat] :
      ( ( ~ ( member_nat @ A @ B )
       => ( A = B2 ) )
     => ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_4475_insertCI,axiom,
    ! [A: int,B: set_int,B2: int] :
      ( ( ~ ( member_int @ A @ B )
       => ( A = B2 ) )
     => ( member_int @ A @ ( insert_int @ B2 @ B ) ) ) ).

% insertCI
thf(fact_4476_Diff__idemp,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ B )
      = ( minus_minus_set_nat @ A2 @ B ) ) ).

% Diff_idemp
thf(fact_4477_Diff__iff,axiom,
    ! [C: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A2 @ B ) )
      = ( ( member_Extended_enat @ C @ A2 )
        & ~ ( member_Extended_enat @ C @ B ) ) ) ).

% Diff_iff
thf(fact_4478_Diff__iff,axiom,
    ! [C: real,A2: set_real,B: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B ) )
      = ( ( member_real @ C @ A2 )
        & ~ ( member_real @ C @ B ) ) ) ).

% Diff_iff
thf(fact_4479_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
      = ( ( member_set_nat @ C @ A2 )
        & ~ ( member_set_nat @ C @ B ) ) ) ).

% Diff_iff
thf(fact_4480_Diff__iff,axiom,
    ! [C: int,A2: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B ) )
      = ( ( member_int @ C @ A2 )
        & ~ ( member_int @ C @ B ) ) ) ).

% Diff_iff
thf(fact_4481_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
      = ( ( member_nat @ C @ A2 )
        & ~ ( member_nat @ C @ B ) ) ) ).

% Diff_iff
thf(fact_4482_DiffI,axiom,
    ! [C: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ A2 )
     => ( ~ ( member_Extended_enat @ C @ B )
       => ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A2 @ B ) ) ) ) ).

% DiffI
thf(fact_4483_DiffI,axiom,
    ! [C: real,A2: set_real,B: set_real] :
      ( ( member_real @ C @ A2 )
     => ( ~ ( member_real @ C @ B )
       => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B ) ) ) ) ).

% DiffI
thf(fact_4484_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ( member_set_nat @ C @ A2 )
     => ( ~ ( member_set_nat @ C @ B )
       => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ) ).

% DiffI
thf(fact_4485_DiffI,axiom,
    ! [C: int,A2: set_int,B: set_int] :
      ( ( member_int @ C @ A2 )
     => ( ~ ( member_int @ C @ B )
       => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B ) ) ) ) ).

% DiffI
thf(fact_4486_DiffI,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( ~ ( member_nat @ C @ B )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).

% DiffI
thf(fact_4487_ComplI,axiom,
    ! [C: extended_enat,A2: set_Extended_enat] :
      ( ~ ( member_Extended_enat @ C @ A2 )
     => ( member_Extended_enat @ C @ ( uminus417252749190364093d_enat @ A2 ) ) ) ).

% ComplI
thf(fact_4488_ComplI,axiom,
    ! [C: real,A2: set_real] :
      ( ~ ( member_real @ C @ A2 )
     => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).

% ComplI
thf(fact_4489_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ ( member_set_nat @ C @ A2 )
     => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).

% ComplI
thf(fact_4490_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ ( member_nat @ C @ A2 )
     => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).

% ComplI
thf(fact_4491_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ ( member_int @ C @ A2 )
     => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% ComplI
thf(fact_4492_Compl__iff,axiom,
    ! [C: extended_enat,A2: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ ( uminus417252749190364093d_enat @ A2 ) )
      = ( ~ ( member_Extended_enat @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_4493_Compl__iff,axiom,
    ! [C: real,A2: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
      = ( ~ ( member_real @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_4494_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_4495_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( ~ ( member_nat @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_4496_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( ~ ( member_int @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_4497_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_4498_singletonI,axiom,
    ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_4499_singletonI,axiom,
    ! [A: extended_enat] : ( member_Extended_enat @ A @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ).

% singletonI
thf(fact_4500_singletonI,axiom,
    ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_4501_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_4502_singletonI,axiom,
    ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_4503_finite__insert,axiom,
    ! [A: real,A2: set_real] :
      ( ( finite_finite_real @ ( insert_real @ A @ A2 ) )
      = ( finite_finite_real @ A2 ) ) ).

% finite_insert
thf(fact_4504_finite__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
      = ( finite_finite_nat @ A2 ) ) ).

% finite_insert
thf(fact_4505_finite__insert,axiom,
    ! [A: complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) )
      = ( finite3207457112153483333omplex @ A2 ) ) ).

% finite_insert
thf(fact_4506_finite__insert,axiom,
    ! [A: int,A2: set_int] :
      ( ( finite_finite_int @ ( insert_int @ A @ A2 ) )
      = ( finite_finite_int @ A2 ) ) ).

% finite_insert
thf(fact_4507_finite__insert,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ ( insert_Extended_enat @ A @ A2 ) )
      = ( finite4001608067531595151d_enat @ A2 ) ) ).

% finite_insert
thf(fact_4508_insert__subset,axiom,
    ! [X: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ ( insert_Extended_enat @ X @ A2 ) @ B )
      = ( ( member_Extended_enat @ X @ B )
        & ( ord_le7203529160286727270d_enat @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_4509_insert__subset,axiom,
    ! [X: real,A2: set_real,B: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B )
      = ( ( member_real @ X @ B )
        & ( ord_less_eq_set_real @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_4510_insert__subset,axiom,
    ! [X: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A2 ) @ B )
      = ( ( member_set_nat @ X @ B )
        & ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_4511_insert__subset,axiom,
    ! [X: nat,A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B )
      = ( ( member_nat @ X @ B )
        & ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_4512_insert__subset,axiom,
    ! [X: int,A2: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B )
      = ( ( member_int @ X @ B )
        & ( ord_less_eq_set_int @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_4513_insert__Diff1,axiom,
    ! [X: extended_enat,B: set_Extended_enat,A2: set_Extended_enat] :
      ( ( member_Extended_enat @ X @ B )
     => ( ( minus_925952699566721837d_enat @ ( insert_Extended_enat @ X @ A2 ) @ B )
        = ( minus_925952699566721837d_enat @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_4514_insert__Diff1,axiom,
    ! [X: real,B: set_real,A2: set_real] :
      ( ( member_real @ X @ B )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B )
        = ( minus_minus_set_real @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_4515_insert__Diff1,axiom,
    ! [X: set_nat,B: set_set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ X @ B )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B )
        = ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_4516_insert__Diff1,axiom,
    ! [X: int,B: set_int,A2: set_int] :
      ( ( member_int @ X @ B )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B )
        = ( minus_minus_set_int @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_4517_insert__Diff1,axiom,
    ! [X: nat,B: set_nat,A2: set_nat] :
      ( ( member_nat @ X @ B )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
        = ( minus_minus_set_nat @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_4518_Diff__insert0,axiom,
    ! [X: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ~ ( member_Extended_enat @ X @ A2 )
     => ( ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ B ) )
        = ( minus_925952699566721837d_enat @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_4519_Diff__insert0,axiom,
    ! [X: real,A2: set_real,B: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B ) )
        = ( minus_minus_set_real @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_4520_Diff__insert0,axiom,
    ! [X: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ B ) )
        = ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_4521_Diff__insert0,axiom,
    ! [X: int,A2: set_int,B: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B ) )
        = ( minus_minus_set_int @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_4522_Diff__insert0,axiom,
    ! [X: nat,A2: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B ) )
        = ( minus_minus_set_nat @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_4523_singleton__conv,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : X4 = A )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv
thf(fact_4524_singleton__conv,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] : X4 = A )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_4525_singleton__conv,axiom,
    ! [A: extended_enat] :
      ( ( collec4429806609662206161d_enat
        @ ^ [X4: extended_enat] : X4 = A )
      = ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ).

% singleton_conv
thf(fact_4526_singleton__conv,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ^ [X4: real] : X4 = A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_4527_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : X4 = A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_4528_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X4: int] : X4 = A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_4529_singleton__conv2,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ( ^ [Y4: list_nat,Z2: list_nat] : Y4 = Z2
          @ A ) )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv2
thf(fact_4530_singleton__conv2,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2
          @ A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv2
thf(fact_4531_singleton__conv2,axiom,
    ! [A: extended_enat] :
      ( ( collec4429806609662206161d_enat
        @ ( ^ [Y4: extended_enat,Z2: extended_enat] : Y4 = Z2
          @ A ) )
      = ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ).

% singleton_conv2
thf(fact_4532_singleton__conv2,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ( ^ [Y4: real,Z2: real] : Y4 = Z2
          @ A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_4533_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y4: nat,Z2: nat] : Y4 = Z2
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_4534_singleton__conv2,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ( ^ [Y4: int,Z2: int] : Y4 = Z2
          @ A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_4535_finite__interval__int1,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B2 ) ) ) ) ).

% finite_interval_int1
thf(fact_4536_singleton__insert__inj__eq_H,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B2: extended_enat] :
      ( ( ( insert_Extended_enat @ A @ A2 )
        = ( insert_Extended_enat @ B2 @ bot_bo7653980558646680370d_enat ) )
      = ( ( A = B2 )
        & ( ord_le7203529160286727270d_enat @ A2 @ ( insert_Extended_enat @ B2 @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4537_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B2: real] :
      ( ( ( insert_real @ A @ A2 )
        = ( insert_real @ B2 @ bot_bot_set_real ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4538_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B2: nat] :
      ( ( ( insert_nat @ A @ A2 )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4539_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B2: int] :
      ( ( ( insert_int @ A @ A2 )
        = ( insert_int @ B2 @ bot_bot_set_int ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B2 @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4540_singleton__insert__inj__eq,axiom,
    ! [B2: extended_enat,A: extended_enat,A2: set_Extended_enat] :
      ( ( ( insert_Extended_enat @ B2 @ bot_bo7653980558646680370d_enat )
        = ( insert_Extended_enat @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_le7203529160286727270d_enat @ A2 @ ( insert_Extended_enat @ B2 @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4541_singleton__insert__inj__eq,axiom,
    ! [B2: real,A: real,A2: set_real] :
      ( ( ( insert_real @ B2 @ bot_bot_set_real )
        = ( insert_real @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4542_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A: nat,A2: set_nat] :
      ( ( ( insert_nat @ B2 @ bot_bot_set_nat )
        = ( insert_nat @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4543_singleton__insert__inj__eq,axiom,
    ! [B2: int,A: int,A2: set_int] :
      ( ( ( insert_int @ B2 @ bot_bot_set_int )
        = ( insert_int @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B2 @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4544_atLeastAtMost__singleton__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ( set_or5403411693681687835d_enat @ A @ B2 )
        = ( insert_Extended_enat @ C @ bot_bo7653980558646680370d_enat ) )
      = ( ( A = B2 )
        & ( B2 = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4545_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B2 )
        = ( insert_nat @ C @ bot_bot_set_nat ) )
      = ( ( A = B2 )
        & ( B2 = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4546_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B2 )
        = ( insert_int @ C @ bot_bot_set_int ) )
      = ( ( A = B2 )
        & ( B2 = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4547_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B2 )
        = ( insert_real @ C @ bot_bot_set_real ) )
      = ( ( A = B2 )
        & ( B2 = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4548_atLeastAtMost__singleton,axiom,
    ! [A: extended_enat] :
      ( ( set_or5403411693681687835d_enat @ A @ A )
      = ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ).

% atLeastAtMost_singleton
thf(fact_4549_atLeastAtMost__singleton,axiom,
    ! [A: nat] :
      ( ( set_or1269000886237332187st_nat @ A @ A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_4550_atLeastAtMost__singleton,axiom,
    ! [A: int] :
      ( ( set_or1266510415728281911st_int @ A @ A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_4551_atLeastAtMost__singleton,axiom,
    ! [A: real] :
      ( ( set_or1222579329274155063t_real @ A @ A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% atLeastAtMost_singleton
thf(fact_4552_insert__Diff__single,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( ( insert_Extended_enat @ A @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
      = ( insert_Extended_enat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4553_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_4554_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_4555_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_4556_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B: set_real] :
      ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B ) ) )
      = ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B ) ) ) ).

% finite_Diff_insert
thf(fact_4557_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B: set_complex] :
      ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B ) ) )
      = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B ) ) ) ).

% finite_Diff_insert
thf(fact_4558_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B: set_int] :
      ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B ) ) )
      = ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B ) ) ) ).

% finite_Diff_insert
thf(fact_4559_finite__Diff__insert,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ B ) ) )
      = ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A2 @ B ) ) ) ).

% finite_Diff_insert
thf(fact_4560_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).

% finite_Diff_insert
thf(fact_4561_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_4562_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N2 ) )
      = ( ( pred_numeral @ K )
        = N2 ) ) ).

% eq_numeral_Suc
thf(fact_4563_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ( suc @ N2 )
        = ( numeral_numeral_nat @ K ) )
      = ( N2
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_4564_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat @ B2 @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4565_subset__Compl__singleton,axiom,
    ! [A2: set_Extended_enat,B2: extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ ( uminus417252749190364093d_enat @ ( insert_Extended_enat @ B2 @ bot_bo7653980558646680370d_enat ) ) )
      = ( ~ ( member_Extended_enat @ B2 @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4566_subset__Compl__singleton,axiom,
    ! [A2: set_real,B2: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B2 @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B2 @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4567_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B2 @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4568_subset__Compl__singleton,axiom,
    ! [A2: set_int,B2: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B2 @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B2 @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4569_less__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% less_numeral_Suc
thf(fact_4570_less__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_4571_le__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% le_numeral_Suc
thf(fact_4572_le__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_4573_diff__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_4574_diff__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% diff_numeral_Suc
thf(fact_4575_max__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_4576_max__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% max_numeral_Suc
thf(fact_4577_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique5052692396658037445od_int @ M2 @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M2 ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_4578_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique5055182867167087721od_nat @ M2 @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M2 ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_4579_set__replicate,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X ) )
        = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_4580_set__replicate,axiom,
    ! [N2: nat,X: extended_enat] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_Extended_enat2 @ ( replic7216382294607269926d_enat @ N2 @ X ) )
        = ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ).

% set_replicate
thf(fact_4581_set__replicate,axiom,
    ! [N2: nat,X: real] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N2 @ X ) )
        = ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_4582_set__replicate,axiom,
    ! [N2: nat,X: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
        = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_4583_set__replicate,axiom,
    ! [N2: nat,X: int] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N2 @ X ) )
        = ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_4584_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_4585_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_4586_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4587_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4588_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R4: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M2 @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_4589_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R4: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R4 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M2 @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_4590_set__diff__eq,axiom,
    ( minus_925952699566721837d_enat
    = ( ^ [A5: set_Extended_enat,B5: set_Extended_enat] :
          ( collec4429806609662206161d_enat
          @ ^ [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A5 )
              & ~ ( member_Extended_enat @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4591_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A5 )
              & ~ ( member_real @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4592_set__diff__eq,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A5 )
              & ~ ( member_list_nat @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4593_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A5 )
              & ~ ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4594_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A5 )
              & ~ ( member_int @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4595_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A5 )
              & ~ ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4596_minus__set__def,axiom,
    ( minus_925952699566721837d_enat
    = ( ^ [A5: set_Extended_enat,B5: set_Extended_enat] :
          ( collec4429806609662206161d_enat
          @ ( minus_2020553357622893040enat_o
            @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ A5 )
            @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_4597_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X4: real] : ( member_real @ X4 @ A5 )
            @ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_4598_minus__set__def,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ( minus_1139252259498527702_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A5 )
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_4599_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 )
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_4600_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A5 )
            @ ^ [X4: int] : ( member_int @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_4601_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_4602_insert__Diff__if,axiom,
    ! [X: extended_enat,B: set_Extended_enat,A2: set_Extended_enat] :
      ( ( ( member_Extended_enat @ X @ B )
       => ( ( minus_925952699566721837d_enat @ ( insert_Extended_enat @ X @ A2 ) @ B )
          = ( minus_925952699566721837d_enat @ A2 @ B ) ) )
      & ( ~ ( member_Extended_enat @ X @ B )
       => ( ( minus_925952699566721837d_enat @ ( insert_Extended_enat @ X @ A2 ) @ B )
          = ( insert_Extended_enat @ X @ ( minus_925952699566721837d_enat @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4603_insert__Diff__if,axiom,
    ! [X: real,B: set_real,A2: set_real] :
      ( ( ( member_real @ X @ B )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B )
          = ( minus_minus_set_real @ A2 @ B ) ) )
      & ( ~ ( member_real @ X @ B )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B )
          = ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4604_insert__Diff__if,axiom,
    ! [X: set_nat,B: set_set_nat,A2: set_set_nat] :
      ( ( ( member_set_nat @ X @ B )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B )
          = ( minus_2163939370556025621et_nat @ A2 @ B ) ) )
      & ( ~ ( member_set_nat @ X @ B )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B )
          = ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4605_insert__Diff__if,axiom,
    ! [X: int,B: set_int,A2: set_int] :
      ( ( ( member_int @ X @ B )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B )
          = ( minus_minus_set_int @ A2 @ B ) ) )
      & ( ~ ( member_int @ X @ B )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B )
          = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4606_insert__Diff__if,axiom,
    ! [X: nat,B: set_nat,A2: set_nat] :
      ( ( ( member_nat @ X @ B )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
          = ( minus_minus_set_nat @ A2 @ B ) ) )
      & ( ~ ( member_nat @ X @ B )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
          = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4607_DiffD2,axiom,
    ! [C: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A2 @ B ) )
     => ~ ( member_Extended_enat @ C @ B ) ) ).

% DiffD2
thf(fact_4608_DiffD2,axiom,
    ! [C: real,A2: set_real,B: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B ) )
     => ~ ( member_real @ C @ B ) ) ).

% DiffD2
thf(fact_4609_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
     => ~ ( member_set_nat @ C @ B ) ) ).

% DiffD2
thf(fact_4610_DiffD2,axiom,
    ! [C: int,A2: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B ) )
     => ~ ( member_int @ C @ B ) ) ).

% DiffD2
thf(fact_4611_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
     => ~ ( member_nat @ C @ B ) ) ).

% DiffD2
thf(fact_4612_DiffD1,axiom,
    ! [C: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A2 @ B ) )
     => ( member_Extended_enat @ C @ A2 ) ) ).

% DiffD1
thf(fact_4613_DiffD1,axiom,
    ! [C: real,A2: set_real,B: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B ) )
     => ( member_real @ C @ A2 ) ) ).

% DiffD1
thf(fact_4614_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
     => ( member_set_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_4615_DiffD1,axiom,
    ! [C: int,A2: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B ) )
     => ( member_int @ C @ A2 ) ) ).

% DiffD1
thf(fact_4616_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
     => ( member_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_4617_DiffE,axiom,
    ! [C: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ ( minus_925952699566721837d_enat @ A2 @ B ) )
     => ~ ( ( member_Extended_enat @ C @ A2 )
         => ( member_Extended_enat @ C @ B ) ) ) ).

% DiffE
thf(fact_4618_DiffE,axiom,
    ! [C: real,A2: set_real,B: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B ) )
     => ~ ( ( member_real @ C @ A2 )
         => ( member_real @ C @ B ) ) ) ).

% DiffE
thf(fact_4619_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
     => ~ ( ( member_set_nat @ C @ A2 )
         => ( member_set_nat @ C @ B ) ) ) ).

% DiffE
thf(fact_4620_DiffE,axiom,
    ! [C: int,A2: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B ) )
     => ~ ( ( member_int @ C @ A2 )
         => ( member_int @ C @ B ) ) ) ).

% DiffE
thf(fact_4621_DiffE,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
     => ~ ( ( member_nat @ C @ A2 )
         => ( member_nat @ C @ B ) ) ) ).

% DiffE
thf(fact_4622_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I4: int] :
            ( ( ord_less_eq_int @ K @ I4 )
           => ( ( P @ I4 )
             => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
       => ( ! [I4: int] :
              ( ( ord_less_eq_int @ I4 @ K )
             => ( ( P @ I4 )
               => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_4623_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I4: int] :
              ( ( ord_less_eq_int @ I4 @ K )
             => ( ( P @ I4 )
               => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_4624_zdvd__zdiffD,axiom,
    ! [K: int,M2: int,N2: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M2 @ N2 ) )
     => ( ( dvd_dvd_int @ K @ N2 )
       => ( dvd_dvd_int @ K @ M2 ) ) ) ).

% zdvd_zdiffD
thf(fact_4625_ComplD,axiom,
    ! [C: extended_enat,A2: set_Extended_enat] :
      ( ( member_Extended_enat @ C @ ( uminus417252749190364093d_enat @ A2 ) )
     => ~ ( member_Extended_enat @ C @ A2 ) ) ).

% ComplD
thf(fact_4626_ComplD,axiom,
    ! [C: real,A2: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
     => ~ ( member_real @ C @ A2 ) ) ).

% ComplD
thf(fact_4627_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
     => ~ ( member_set_nat @ C @ A2 ) ) ).

% ComplD
thf(fact_4628_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
     => ~ ( member_nat @ C @ A2 ) ) ).

% ComplD
thf(fact_4629_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
     => ~ ( member_int @ C @ A2 ) ) ).

% ComplD
thf(fact_4630_uminus__set__def,axiom,
    ( uminus417252749190364093d_enat
    = ( ^ [A5: set_Extended_enat] :
          ( collec4429806609662206161d_enat
          @ ( uminus6636779312473996640enat_o
            @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_4631_uminus__set__def,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A5: set_real] :
          ( collect_real
          @ ( uminus_uminus_real_o
            @ ^ [X4: real] : ( member_real @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_4632_uminus__set__def,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A5: set_list_nat] :
          ( collect_list_nat
          @ ( uminus5770388063884162150_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_4633_uminus__set__def,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A5: set_set_nat] :
          ( collect_set_nat
          @ ( uminus6401447641752708672_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_4634_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A5: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_4635_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A5: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_4636_Collect__neg__eq,axiom,
    ! [P: real > $o] :
      ( ( collect_real
        @ ^ [X4: real] :
            ~ ( P @ X4 ) )
      = ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).

% Collect_neg_eq
thf(fact_4637_Collect__neg__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ~ ( P @ X4 ) )
      = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_4638_Collect__neg__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ~ ( P @ X4 ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_4639_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ~ ( P @ X4 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_4640_Collect__neg__eq,axiom,
    ! [P: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ~ ( P @ X4 ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_4641_Compl__eq,axiom,
    ( uminus417252749190364093d_enat
    = ( ^ [A5: set_Extended_enat] :
          ( collec4429806609662206161d_enat
          @ ^ [X4: extended_enat] :
              ~ ( member_Extended_enat @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_4642_Compl__eq,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A5: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ~ ( member_real @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_4643_Compl__eq,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ~ ( member_list_nat @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_4644_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ~ ( member_set_nat @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_4645_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ~ ( member_nat @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_4646_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A5: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ~ ( member_int @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_4647_insert__compr,axiom,
    ( insert_Extended_enat
    = ( ^ [A3: extended_enat,B5: set_Extended_enat] :
          ( collec4429806609662206161d_enat
          @ ^ [X4: extended_enat] :
              ( ( X4 = A3 )
              | ( member_Extended_enat @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_4648_insert__compr,axiom,
    ( insert_real
    = ( ^ [A3: real,B5: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ( ( X4 = A3 )
              | ( member_real @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_4649_insert__compr,axiom,
    ( insert_list_nat
    = ( ^ [A3: list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( X4 = A3 )
              | ( member_list_nat @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_4650_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A3: set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( X4 = A3 )
              | ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_4651_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A3: nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( X4 = A3 )
              | ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_4652_insert__compr,axiom,
    ( insert_int
    = ( ^ [A3: int,B5: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( X4 = A3 )
              | ( member_int @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_4653_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_4654_insert__Collect,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( insert_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( collect_list_nat
        @ ^ [U2: list_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4655_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_4656_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_4657_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_4658_mk__disjoint__insert,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( ( member_Extended_enat @ A @ A2 )
     => ? [B8: set_Extended_enat] :
          ( ( A2
            = ( insert_Extended_enat @ A @ B8 ) )
          & ~ ( member_Extended_enat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_4659_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_4660_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_4661_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_4662_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_4663_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_4664_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_4665_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_4666_insert__eq__iff,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,B2: extended_enat,B: set_Extended_enat] :
      ( ~ ( member_Extended_enat @ A @ A2 )
     => ( ~ ( member_Extended_enat @ B2 @ B )
       => ( ( ( insert_Extended_enat @ A @ A2 )
            = ( insert_Extended_enat @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C5: set_Extended_enat] :
                  ( ( A2
                    = ( insert_Extended_enat @ B2 @ C5 ) )
                  & ~ ( member_Extended_enat @ B2 @ C5 )
                  & ( B
                    = ( insert_Extended_enat @ A @ C5 ) )
                  & ~ ( member_Extended_enat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4667_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B2: real,B: set_real] :
      ( ~ ( member_real @ A @ A2 )
     => ( ~ ( member_real @ B2 @ B )
       => ( ( ( insert_real @ A @ A2 )
            = ( insert_real @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C5: set_real] :
                  ( ( A2
                    = ( insert_real @ B2 @ C5 ) )
                  & ~ ( member_real @ B2 @ C5 )
                  & ( B
                    = ( insert_real @ A @ C5 ) )
                  & ~ ( member_real @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4668_insert__eq__iff,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_nat,B: set_set_nat] :
      ( ~ ( member_set_nat @ A @ A2 )
     => ( ~ ( member_set_nat @ B2 @ B )
       => ( ( ( insert_set_nat @ A @ A2 )
            = ( insert_set_nat @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C5: set_set_nat] :
                  ( ( A2
                    = ( insert_set_nat @ B2 @ C5 ) )
                  & ~ ( member_set_nat @ B2 @ C5 )
                  & ( B
                    = ( insert_set_nat @ A @ C5 ) )
                  & ~ ( member_set_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4669_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
      ( ~ ( member_nat @ A @ A2 )
     => ( ~ ( member_nat @ B2 @ B )
       => ( ( ( insert_nat @ A @ A2 )
            = ( insert_nat @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C5: set_nat] :
                  ( ( A2
                    = ( insert_nat @ B2 @ C5 ) )
                  & ~ ( member_nat @ B2 @ C5 )
                  & ( B
                    = ( insert_nat @ A @ C5 ) )
                  & ~ ( member_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4670_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B2: int,B: set_int] :
      ( ~ ( member_int @ A @ A2 )
     => ( ~ ( member_int @ B2 @ B )
       => ( ( ( insert_int @ A @ A2 )
            = ( insert_int @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C5: set_int] :
                  ( ( A2
                    = ( insert_int @ B2 @ C5 ) )
                  & ~ ( member_int @ B2 @ C5 )
                  & ( B
                    = ( insert_int @ A @ C5 ) )
                  & ~ ( member_int @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4671_insert__absorb,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( ( member_Extended_enat @ A @ A2 )
     => ( ( insert_Extended_enat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4672_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ( insert_real @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4673_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_4674_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4675_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4676_insert__ident,axiom,
    ! [X: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ~ ( member_Extended_enat @ X @ A2 )
     => ( ~ ( member_Extended_enat @ X @ B )
       => ( ( ( insert_Extended_enat @ X @ A2 )
            = ( insert_Extended_enat @ X @ B ) )
          = ( A2 = B ) ) ) ) ).

% insert_ident
thf(fact_4677_insert__ident,axiom,
    ! [X: real,A2: set_real,B: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ~ ( member_real @ X @ B )
       => ( ( ( insert_real @ X @ A2 )
            = ( insert_real @ X @ B ) )
          = ( A2 = B ) ) ) ) ).

% insert_ident
thf(fact_4678_insert__ident,axiom,
    ! [X: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ~ ( member_set_nat @ X @ B )
       => ( ( ( insert_set_nat @ X @ A2 )
            = ( insert_set_nat @ X @ B ) )
          = ( A2 = B ) ) ) ) ).

% insert_ident
thf(fact_4679_insert__ident,axiom,
    ! [X: nat,A2: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ~ ( member_nat @ X @ B )
       => ( ( ( insert_nat @ X @ A2 )
            = ( insert_nat @ X @ B ) )
          = ( A2 = B ) ) ) ) ).

% insert_ident
thf(fact_4680_insert__ident,axiom,
    ! [X: int,A2: set_int,B: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ~ ( member_int @ X @ B )
       => ( ( ( insert_int @ X @ A2 )
            = ( insert_int @ X @ B ) )
          = ( A2 = B ) ) ) ) ).

% insert_ident
thf(fact_4681_Set_Oset__insert,axiom,
    ! [X: extended_enat,A2: set_Extended_enat] :
      ( ( member_Extended_enat @ X @ A2 )
     => ~ ! [B8: set_Extended_enat] :
            ( ( A2
              = ( insert_Extended_enat @ X @ B8 ) )
           => ( member_Extended_enat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_4682_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_4683_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_4684_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_4685_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_4686_insertI2,axiom,
    ! [A: extended_enat,B: set_Extended_enat,B2: extended_enat] :
      ( ( member_Extended_enat @ A @ B )
     => ( member_Extended_enat @ A @ ( insert_Extended_enat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_4687_insertI2,axiom,
    ! [A: real,B: set_real,B2: real] :
      ( ( member_real @ A @ B )
     => ( member_real @ A @ ( insert_real @ B2 @ B ) ) ) ).

% insertI2
thf(fact_4688_insertI2,axiom,
    ! [A: set_nat,B: set_set_nat,B2: set_nat] :
      ( ( member_set_nat @ A @ B )
     => ( member_set_nat @ A @ ( insert_set_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_4689_insertI2,axiom,
    ! [A: nat,B: set_nat,B2: nat] :
      ( ( member_nat @ A @ B )
     => ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_4690_insertI2,axiom,
    ! [A: int,B: set_int,B2: int] :
      ( ( member_int @ A @ B )
     => ( member_int @ A @ ( insert_int @ B2 @ B ) ) ) ).

% insertI2
thf(fact_4691_insertI1,axiom,
    ! [A: extended_enat,B: set_Extended_enat] : ( member_Extended_enat @ A @ ( insert_Extended_enat @ A @ B ) ) ).

% insertI1
thf(fact_4692_insertI1,axiom,
    ! [A: real,B: set_real] : ( member_real @ A @ ( insert_real @ A @ B ) ) ).

% insertI1
thf(fact_4693_insertI1,axiom,
    ! [A: set_nat,B: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B ) ) ).

% insertI1
thf(fact_4694_insertI1,axiom,
    ! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).

% insertI1
thf(fact_4695_insertI1,axiom,
    ! [A: int,B: set_int] : ( member_int @ A @ ( insert_int @ A @ B ) ) ).

% insertI1
thf(fact_4696_insertE,axiom,
    ! [A: extended_enat,B2: extended_enat,A2: set_Extended_enat] :
      ( ( member_Extended_enat @ A @ ( insert_Extended_enat @ B2 @ A2 ) )
     => ( ( A != B2 )
       => ( member_Extended_enat @ A @ A2 ) ) ) ).

% insertE
thf(fact_4697_insertE,axiom,
    ! [A: real,B2: real,A2: set_real] :
      ( ( member_real @ A @ ( insert_real @ B2 @ A2 ) )
     => ( ( A != B2 )
       => ( member_real @ A @ A2 ) ) ) ).

% insertE
thf(fact_4698_insertE,axiom,
    ! [A: set_nat,B2: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B2 @ A2 ) )
     => ( ( A != B2 )
       => ( member_set_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_4699_insertE,axiom,
    ! [A: nat,B2: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
     => ( ( A != B2 )
       => ( member_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_4700_insertE,axiom,
    ! [A: int,B2: int,A2: set_int] :
      ( ( member_int @ A @ ( insert_int @ B2 @ A2 ) )
     => ( ( A != B2 )
       => ( member_int @ A @ A2 ) ) ) ).

% insertE
thf(fact_4701_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I4: int] :
              ( ( ord_less_eq_int @ K @ I4 )
             => ( ( P @ I4 )
               => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_4702_singletonD,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( member_set_nat @ B2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_4703_singletonD,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( member_Extended_enat @ B2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_4704_singletonD,axiom,
    ! [B2: real,A: real] :
      ( ( member_real @ B2 @ ( insert_real @ A @ bot_bot_set_real ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_4705_singletonD,axiom,
    ! [B2: nat,A: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_4706_singletonD,axiom,
    ! [B2: int,A: int] :
      ( ( member_int @ B2 @ ( insert_int @ A @ bot_bot_set_int ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_4707_singleton__iff,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( member_set_nat @ B2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_4708_singleton__iff,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( member_Extended_enat @ B2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_4709_singleton__iff,axiom,
    ! [B2: real,A: real] :
      ( ( member_real @ B2 @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_4710_singleton__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_4711_singleton__iff,axiom,
    ! [B2: int,A: int] :
      ( ( member_int @ B2 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_4712_doubleton__eq__iff,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ( insert_Extended_enat @ A @ ( insert_Extended_enat @ B2 @ bot_bo7653980558646680370d_enat ) )
        = ( insert_Extended_enat @ C @ ( insert_Extended_enat @ D @ bot_bo7653980558646680370d_enat ) ) )
      = ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4713_doubleton__eq__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( insert_real @ A @ ( insert_real @ B2 @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4714_doubleton__eq__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4715_doubleton__eq__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( insert_int @ A @ ( insert_int @ B2 @ bot_bot_set_int ) )
        = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
      = ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4716_insert__not__empty,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( ( insert_Extended_enat @ A @ A2 )
     != bot_bo7653980558646680370d_enat ) ).

% insert_not_empty
thf(fact_4717_insert__not__empty,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real @ A @ A2 )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_4718_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ A2 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_4719_insert__not__empty,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ A2 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_4720_singleton__inject,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat )
        = ( insert_Extended_enat @ B2 @ bot_bo7653980558646680370d_enat ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_4721_singleton__inject,axiom,
    ! [A: real,B2: real] :
      ( ( ( insert_real @ A @ bot_bot_set_real )
        = ( insert_real @ B2 @ bot_bot_set_real ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_4722_singleton__inject,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_4723_singleton__inject,axiom,
    ! [A: int,B2: int] :
      ( ( ( insert_int @ A @ bot_bot_set_int )
        = ( insert_int @ B2 @ bot_bot_set_int ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_4724_Compl__insert,axiom,
    ! [X: extended_enat,A2: set_Extended_enat] :
      ( ( uminus417252749190364093d_enat @ ( insert_Extended_enat @ X @ A2 ) )
      = ( minus_925952699566721837d_enat @ ( uminus417252749190364093d_enat @ A2 ) @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ).

% Compl_insert
thf(fact_4725_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_4726_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_4727_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_4728_Diff__insert,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ B ) )
      = ( minus_925952699566721837d_enat @ ( minus_925952699566721837d_enat @ A2 @ B ) @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ).

% Diff_insert
thf(fact_4729_Diff__insert,axiom,
    ! [A2: set_real,A: real,B: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_4730_Diff__insert,axiom,
    ! [A2: set_int,A: int,B: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_4731_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_4732_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_4733_insert__Diff,axiom,
    ! [A: extended_enat,A2: set_Extended_enat] :
      ( ( member_Extended_enat @ A @ A2 )
     => ( ( insert_Extended_enat @ A @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4734_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_4735_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_4736_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_4737_Diff__insert2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ B ) )
      = ( minus_925952699566721837d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) @ B ) ) ).

% Diff_insert2
thf(fact_4738_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B ) ) ).

% Diff_insert2
thf(fact_4739_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B ) ) ).

% Diff_insert2
thf(fact_4740_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B ) ) ).

% Diff_insert2
thf(fact_4741_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_4742_Diff__insert__absorb,axiom,
    ! [X: extended_enat,A2: set_Extended_enat] :
      ( ~ ( member_Extended_enat @ X @ A2 )
     => ( ( minus_925952699566721837d_enat @ ( insert_Extended_enat @ X @ A2 ) @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4743_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_4744_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_4745_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_4746_finite_OinsertI,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( finite_finite_real @ ( insert_real @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4747_finite_OinsertI,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4748_finite_OinsertI,axiom,
    ! [A2: set_complex,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4749_finite_OinsertI,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( finite_finite_int @ ( insert_int @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4750_finite_OinsertI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite4001608067531595151d_enat @ ( insert_Extended_enat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4751_insert__mono,axiom,
    ! [C4: set_real,D6: set_real,A: real] :
      ( ( ord_less_eq_set_real @ C4 @ D6 )
     => ( ord_less_eq_set_real @ ( insert_real @ A @ C4 ) @ ( insert_real @ A @ D6 ) ) ) ).

% insert_mono
thf(fact_4752_insert__mono,axiom,
    ! [C4: set_nat,D6: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C4 @ D6 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C4 ) @ ( insert_nat @ A @ D6 ) ) ) ).

% insert_mono
thf(fact_4753_insert__mono,axiom,
    ! [C4: set_int,D6: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C4 @ D6 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C4 ) @ ( insert_int @ A @ D6 ) ) ) ).

% insert_mono
thf(fact_4754_subset__insert,axiom,
    ! [X: extended_enat,A2: set_Extended_enat,B: set_Extended_enat] :
      ( ~ ( member_Extended_enat @ X @ A2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ ( insert_Extended_enat @ X @ B ) )
        = ( ord_le7203529160286727270d_enat @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_4755_subset__insert,axiom,
    ! [X: real,A2: set_real,B: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B ) )
        = ( ord_less_eq_set_real @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_4756_subset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat,B: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B ) )
        = ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_4757_subset__insert,axiom,
    ! [X: nat,A2: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) )
        = ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_4758_subset__insert,axiom,
    ! [X: int,A2: set_int,B: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B ) )
        = ( ord_less_eq_set_int @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_4759_subset__insertI,axiom,
    ! [B: set_real,A: real] : ( ord_less_eq_set_real @ B @ ( insert_real @ A @ B ) ) ).

% subset_insertI
thf(fact_4760_subset__insertI,axiom,
    ! [B: set_nat,A: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A @ B ) ) ).

% subset_insertI
thf(fact_4761_subset__insertI,axiom,
    ! [B: set_int,A: int] : ( ord_less_eq_set_int @ B @ ( insert_int @ A @ B ) ) ).

% subset_insertI
thf(fact_4762_subset__insertI2,axiom,
    ! [A2: set_real,B: set_real,B2: real] :
      ( ( ord_less_eq_set_real @ A2 @ B )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_4763_subset__insertI2,axiom,
    ! [A2: set_nat,B: set_nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_4764_subset__insertI2,axiom,
    ! [A2: set_int,B: set_int,B2: int] :
      ( ( ord_less_eq_set_int @ A2 @ B )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_4765_insert__subsetI,axiom,
    ! [X: extended_enat,A2: set_Extended_enat,X8: set_Extended_enat] :
      ( ( member_Extended_enat @ X @ A2 )
     => ( ( ord_le7203529160286727270d_enat @ X8 @ A2 )
       => ( ord_le7203529160286727270d_enat @ ( insert_Extended_enat @ X @ X8 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4766_insert__subsetI,axiom,
    ! [X: real,A2: set_real,X8: set_real] :
      ( ( member_real @ X @ A2 )
     => ( ( ord_less_eq_set_real @ X8 @ A2 )
       => ( ord_less_eq_set_real @ ( insert_real @ X @ X8 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4767_insert__subsetI,axiom,
    ! [X: set_nat,A2: set_set_nat,X8: set_set_nat] :
      ( ( member_set_nat @ X @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ X8 @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ X8 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4768_insert__subsetI,axiom,
    ! [X: nat,A2: set_nat,X8: set_nat] :
      ( ( member_nat @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ X8 @ A2 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X @ X8 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4769_insert__subsetI,axiom,
    ! [X: int,A2: set_int,X8: set_int] :
      ( ( member_int @ X @ A2 )
     => ( ( ord_less_eq_set_int @ X8 @ A2 )
       => ( ord_less_eq_set_int @ ( insert_int @ X @ X8 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4770_subset__Diff__insert,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,X: extended_enat,C4: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ ( minus_925952699566721837d_enat @ B @ ( insert_Extended_enat @ X @ C4 ) ) )
      = ( ( ord_le7203529160286727270d_enat @ A2 @ ( minus_925952699566721837d_enat @ B @ C4 ) )
        & ~ ( member_Extended_enat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4771_subset__Diff__insert,axiom,
    ! [A2: set_real,B: set_real,X: real,C4: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B @ ( insert_real @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B @ C4 ) )
        & ~ ( member_real @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4772_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B: set_set_nat,X: set_nat,C4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B @ ( insert_set_nat @ X @ C4 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B @ C4 ) )
        & ~ ( member_set_nat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4773_subset__Diff__insert,axiom,
    ! [A2: set_nat,B: set_nat,X: nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C4 ) )
        & ~ ( member_nat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4774_subset__Diff__insert,axiom,
    ! [A2: set_int,B: set_int,X: int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B @ ( insert_int @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B @ C4 ) )
        & ~ ( member_int @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4775_Collect__conv__if,axiom,
    ! [P: list_nat > $o,A: list_nat] :
      ( ( ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
thf(fact_4776_Collect__conv__if,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_4777_Collect__conv__if,axiom,
    ! [P: extended_enat > $o,A: extended_enat] :
      ( ( ( P @ A )
       => ( ( collec4429806609662206161d_enat
            @ ^ [X4: extended_enat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec4429806609662206161d_enat
            @ ^ [X4: extended_enat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bo7653980558646680370d_enat ) ) ) ).

% Collect_conv_if
thf(fact_4778_Collect__conv__if,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X4: real] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X4: real] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_4779_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_4780_Collect__conv__if,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_4781_Collect__conv__if2,axiom,
    ! [P: list_nat > $o,A: list_nat] :
      ( ( ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
thf(fact_4782_Collect__conv__if2,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_4783_Collect__conv__if2,axiom,
    ! [P: extended_enat > $o,A: extended_enat] :
      ( ( ( P @ A )
       => ( ( collec4429806609662206161d_enat
            @ ^ [X4: extended_enat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec4429806609662206161d_enat
            @ ^ [X4: extended_enat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bo7653980558646680370d_enat ) ) ) ).

% Collect_conv_if2
thf(fact_4784_Collect__conv__if2,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X4: real] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X4: real] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_4785_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_4786_Collect__conv__if2,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_4787_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A6: set_complex] :
              ( ? [A4: complex] :
                  ( A
                  = ( insert_complex @ A4 @ A6 ) )
             => ~ ( finite3207457112153483333omplex @ A6 ) ) ) ) ).

% finite.cases
thf(fact_4788_finite_Ocases,axiom,
    ! [A: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A )
     => ( ( A != bot_bo7653980558646680370d_enat )
       => ~ ! [A6: set_Extended_enat] :
              ( ? [A4: extended_enat] :
                  ( A
                  = ( insert_Extended_enat @ A4 @ A6 ) )
             => ~ ( finite4001608067531595151d_enat @ A6 ) ) ) ) ).

% finite.cases
thf(fact_4789_finite_Ocases,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ~ ! [A6: set_real] :
              ( ? [A4: real] :
                  ( A
                  = ( insert_real @ A4 @ A6 ) )
             => ~ ( finite_finite_real @ A6 ) ) ) ) ).

% finite.cases
thf(fact_4790_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A6: set_nat] :
              ( ? [A4: nat] :
                  ( A
                  = ( insert_nat @ A4 @ A6 ) )
             => ~ ( finite_finite_nat @ A6 ) ) ) ) ).

% finite.cases
thf(fact_4791_finite_Ocases,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ~ ! [A6: set_int] :
              ( ? [A4: int] :
                  ( A
                  = ( insert_int @ A4 @ A6 ) )
             => ~ ( finite_finite_int @ A6 ) ) ) ) ).

% finite.cases
thf(fact_4792_finite_Osimps,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A3: set_complex] :
          ( ( A3 = bot_bot_set_complex )
          | ? [A5: set_complex,B3: complex] :
              ( ( A3
                = ( insert_complex @ B3 @ A5 ) )
              & ( finite3207457112153483333omplex @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4793_finite_Osimps,axiom,
    ( finite4001608067531595151d_enat
    = ( ^ [A3: set_Extended_enat] :
          ( ( A3 = bot_bo7653980558646680370d_enat )
          | ? [A5: set_Extended_enat,B3: extended_enat] :
              ( ( A3
                = ( insert_Extended_enat @ B3 @ A5 ) )
              & ( finite4001608067531595151d_enat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4794_finite_Osimps,axiom,
    ( finite_finite_real
    = ( ^ [A3: set_real] :
          ( ( A3 = bot_bot_set_real )
          | ? [A5: set_real,B3: real] :
              ( ( A3
                = ( insert_real @ B3 @ A5 ) )
              & ( finite_finite_real @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4795_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A3: set_nat] :
          ( ( A3 = bot_bot_set_nat )
          | ? [A5: set_nat,B3: nat] :
              ( ( A3
                = ( insert_nat @ B3 @ A5 ) )
              & ( finite_finite_nat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4796_finite_Osimps,axiom,
    ( finite_finite_int
    = ( ^ [A3: set_int] :
          ( ( A3 = bot_bot_set_int )
          | ? [A5: set_int,B3: int] :
              ( ( A3
                = ( insert_int @ B3 @ A5 ) )
              & ( finite_finite_int @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4797_finite__induct,axiom,
    ! [F3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X5: set_nat,F4: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F4 )
             => ( ~ ( member_set_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4798_finite__induct,axiom,
    ! [F3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,F4: set_complex] :
              ( ( finite3207457112153483333omplex @ F4 )
             => ( ~ ( member_complex @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4799_finite__induct,axiom,
    ! [F3: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,F4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ F4 )
             => ( ~ ( member_Extended_enat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4800_finite__induct,axiom,
    ! [F3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,F4: set_real] :
              ( ( finite_finite_real @ F4 )
             => ( ~ ( member_real @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4801_finite__induct,axiom,
    ! [F3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4802_finite__induct,axiom,
    ! [F3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,F4: set_int] :
              ( ( finite_finite_int @ F4 )
             => ( ~ ( member_int @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4803_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( F3 != bot_bot_set_set_nat )
       => ( ! [X5: set_nat] : ( P @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) )
         => ( ! [X5: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ ( member_set_nat @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4804_finite__ne__induct,axiom,
    ! [F3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X5: complex] : ( P @ ( insert_complex @ X5 @ bot_bot_set_complex ) )
         => ( ! [X5: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ ( member_complex @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4805_finite__ne__induct,axiom,
    ! [F3: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( F3 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] : ( P @ ( insert_Extended_enat @ X5 @ bot_bo7653980558646680370d_enat ) )
         => ( ! [X5: extended_enat,F4: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F4 )
               => ( ( F4 != bot_bo7653980558646680370d_enat )
                 => ( ~ ( member_Extended_enat @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4806_finite__ne__induct,axiom,
    ! [F3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( F3 != bot_bot_set_real )
       => ( ! [X5: real] : ( P @ ( insert_real @ X5 @ bot_bot_set_real ) )
         => ( ! [X5: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ ( member_real @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4807_finite__ne__induct,axiom,
    ! [F3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( F3 != bot_bot_set_nat )
       => ( ! [X5: nat] : ( P @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
         => ( ! [X5: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4808_finite__ne__induct,axiom,
    ! [F3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( F3 != bot_bot_set_int )
       => ( ! [X5: int] : ( P @ ( insert_int @ X5 @ bot_bot_set_int ) )
         => ( ! [X5: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ ( member_int @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4809_infinite__finite__induct,axiom,
    ! [P: set_set_nat > $o,A2: set_set_nat] :
      ( ! [A6: set_set_nat] :
          ( ~ ( finite1152437895449049373et_nat @ A6 )
         => ( P @ A6 ) )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X5: set_nat,F4: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F4 )
             => ( ~ ( member_set_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4810_infinite__finite__induct,axiom,
    ! [P: set_complex > $o,A2: set_complex] :
      ( ! [A6: set_complex] :
          ( ~ ( finite3207457112153483333omplex @ A6 )
         => ( P @ A6 ) )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,F4: set_complex] :
              ( ( finite3207457112153483333omplex @ F4 )
             => ( ~ ( member_complex @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4811_infinite__finite__induct,axiom,
    ! [P: set_Extended_enat > $o,A2: set_Extended_enat] :
      ( ! [A6: set_Extended_enat] :
          ( ~ ( finite4001608067531595151d_enat @ A6 )
         => ( P @ A6 ) )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,F4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ F4 )
             => ( ~ ( member_Extended_enat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4812_infinite__finite__induct,axiom,
    ! [P: set_real > $o,A2: set_real] :
      ( ! [A6: set_real] :
          ( ~ ( finite_finite_real @ A6 )
         => ( P @ A6 ) )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,F4: set_real] :
              ( ( finite_finite_real @ F4 )
             => ( ~ ( member_real @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4813_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A2: set_nat] :
      ( ! [A6: set_nat] :
          ( ~ ( finite_finite_nat @ A6 )
         => ( P @ A6 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4814_infinite__finite__induct,axiom,
    ! [P: set_int > $o,A2: set_int] :
      ( ! [A6: set_int] :
          ( ~ ( finite_finite_int @ A6 )
         => ( P @ A6 ) )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,F4: set_int] :
              ( ( finite_finite_int @ F4 )
             => ( ~ ( member_int @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4815_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_4816_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_4817_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_4818_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_4819_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_4820_infinite__coinduct,axiom,
    ! [X8: set_complex > $o,A2: set_complex] :
      ( ( X8 @ A2 )
     => ( ! [A6: set_complex] :
            ( ( X8 @ A6 )
           => ? [X2: complex] :
                ( ( member_complex @ X2 @ A6 )
                & ( ( X8 @ ( minus_811609699411566653omplex @ A6 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A6 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_4821_infinite__coinduct,axiom,
    ! [X8: set_Extended_enat > $o,A2: set_Extended_enat] :
      ( ( X8 @ A2 )
     => ( ! [A6: set_Extended_enat] :
            ( ( X8 @ A6 )
           => ? [X2: extended_enat] :
                ( ( member_Extended_enat @ X2 @ A6 )
                & ( ( X8 @ ( minus_925952699566721837d_enat @ A6 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) )
                  | ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A6 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
       => ~ ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_4822_infinite__coinduct,axiom,
    ! [X8: set_real > $o,A2: set_real] :
      ( ( X8 @ A2 )
     => ( ! [A6: set_real] :
            ( ( X8 @ A6 )
           => ? [X2: real] :
                ( ( member_real @ X2 @ A6 )
                & ( ( X8 @ ( minus_minus_set_real @ A6 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A6 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_4823_infinite__coinduct,axiom,
    ! [X8: set_int > $o,A2: set_int] :
      ( ( X8 @ A2 )
     => ( ! [A6: set_int] :
            ( ( X8 @ A6 )
           => ? [X2: int] :
                ( ( member_int @ X2 @ A6 )
                & ( ( X8 @ ( minus_minus_set_int @ A6 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A6 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_4824_infinite__coinduct,axiom,
    ! [X8: set_nat > $o,A2: set_nat] :
      ( ( X8 @ A2 )
     => ( ! [A6: set_nat] :
            ( ( X8 @ A6 )
           => ? [X2: nat] :
                ( ( member_nat @ X2 @ A6 )
                & ( ( X8 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_4825_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: set_nat,A6: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A6 )
             => ( ( member_set_nat @ A4 @ A6 )
               => ( ( P @ A6 )
                 => ( P @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_4826_finite__empty__induct,axiom,
    ! [A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: complex,A6: set_complex] :
              ( ( finite3207457112153483333omplex @ A6 )
             => ( ( member_complex @ A4 @ A6 )
               => ( ( P @ A6 )
                 => ( P @ ( minus_811609699411566653omplex @ A6 @ ( insert_complex @ A4 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_4827_finite__empty__induct,axiom,
    ! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: extended_enat,A6: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A6 )
             => ( ( member_Extended_enat @ A4 @ A6 )
               => ( ( P @ A6 )
                 => ( P @ ( minus_925952699566721837d_enat @ A6 @ ( insert_Extended_enat @ A4 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
         => ( P @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% finite_empty_induct
thf(fact_4828_finite__empty__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: real,A6: set_real] :
              ( ( finite_finite_real @ A6 )
             => ( ( member_real @ A4 @ A6 )
               => ( ( P @ A6 )
                 => ( P @ ( minus_minus_set_real @ A6 @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_4829_finite__empty__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: int,A6: set_int] :
              ( ( finite_finite_int @ A6 )
             => ( ( member_int @ A4 @ A6 )
               => ( ( P @ A6 )
                 => ( P @ ( minus_minus_set_int @ A6 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_4830_finite__empty__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: nat,A6: set_nat] :
              ( ( finite_finite_nat @ A6 )
             => ( ( member_nat @ A4 @ A6 )
               => ( ( P @ A6 )
                 => ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_4831_subset__singleton__iff,axiom,
    ! [X8: set_Extended_enat,A: extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ X8 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) )
      = ( ( X8 = bot_bo7653980558646680370d_enat )
        | ( X8
          = ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% subset_singleton_iff
thf(fact_4832_subset__singleton__iff,axiom,
    ! [X8: set_real,A: real] :
      ( ( ord_less_eq_set_real @ X8 @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( ( X8 = bot_bot_set_real )
        | ( X8
          = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_4833_subset__singleton__iff,axiom,
    ! [X8: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X8 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X8 = bot_bot_set_nat )
        | ( X8
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_4834_subset__singleton__iff,axiom,
    ! [X8: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X8 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X8 = bot_bot_set_int )
        | ( X8
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_4835_subset__singletonD,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) )
     => ( ( A2 = bot_bo7653980558646680370d_enat )
        | ( A2
          = ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% subset_singletonD
thf(fact_4836_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_4837_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_4838_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_4839_Diff__single__insert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,B: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) @ B )
     => ( ord_le7203529160286727270d_enat @ A2 @ ( insert_Extended_enat @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_4840_Diff__single__insert,axiom,
    ! [A2: set_real,X: real,B: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_4841_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_4842_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_4843_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B ) )
      = ( ( ( member_set_nat @ X @ A2 )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B ) )
        & ( ~ ( member_set_nat @ X @ A2 )
         => ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_4844_subset__insert__iff,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,B: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ ( insert_Extended_enat @ X @ B ) )
      = ( ( ( member_Extended_enat @ X @ A2 )
         => ( ord_le7203529160286727270d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) @ B ) )
        & ( ~ ( member_Extended_enat @ X @ A2 )
         => ( ord_le7203529160286727270d_enat @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_4845_subset__insert__iff,axiom,
    ! [A2: set_real,X: real,B: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B ) )
      = ( ( ( member_real @ X @ A2 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ord_less_eq_set_real @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_4846_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) )
      = ( ( ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_4847_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B ) )
      = ( ( ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_4848_atLeastAtMost__singleton_H,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( A = B2 )
     => ( ( set_or5403411693681687835d_enat @ A @ B2 )
        = ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4849_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
     => ( ( set_or1269000886237332187st_nat @ A @ B2 )
        = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4850_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
     => ( ( set_or1266510415728281911st_int @ A @ B2 )
        = ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4851_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
     => ( ( set_or1222579329274155063t_real @ A @ B2 )
        = ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4852_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_4853_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: set_complex > $o,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y6: complex] :
                    ( ( member_complex @ Y6 @ S4 )
                   => ( ord_less_eq_real @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4854_finite__ranking__induct,axiom,
    ! [S2: set_Extended_enat,P: set_Extended_enat > $o,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,S4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ S4 )
             => ( ! [Y6: extended_enat] :
                    ( ( member_Extended_enat @ Y6 @ S4 )
                   => ( ord_less_eq_real @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_Extended_enat @ X5 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4855_finite__ranking__induct,axiom,
    ! [S2: set_real,P: set_real > $o,F: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y6: real] :
                    ( ( member_real @ Y6 @ S4 )
                   => ( ord_less_eq_real @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4856_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S4 )
                   => ( ord_less_eq_real @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X5 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4857_finite__ranking__induct,axiom,
    ! [S2: set_int,P: set_int > $o,F: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y6: int] :
                    ( ( member_int @ Y6 @ S4 )
                   => ( ord_less_eq_real @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X5 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4858_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: set_complex > $o,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y6: complex] :
                    ( ( member_complex @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4859_finite__ranking__induct,axiom,
    ! [S2: set_Extended_enat,P: set_Extended_enat > $o,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,S4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ S4 )
             => ( ! [Y6: extended_enat] :
                    ( ( member_Extended_enat @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_Extended_enat @ X5 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4860_finite__ranking__induct,axiom,
    ! [S2: set_real,P: set_real > $o,F: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y6: real] :
                    ( ( member_real @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4861_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X5 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4862_finite__ranking__induct,axiom,
    ! [S2: set_int,P: set_int > $o,F: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y6: int] :
                    ( ( member_int @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X5 @ S4 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4863_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B4: nat,A6: set_nat] :
              ( ( finite_finite_nat @ A6 )
             => ( ! [X2: nat] :
                    ( ( member_nat @ X2 @ A6 )
                   => ( ord_less_nat @ X2 @ B4 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_nat @ B4 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4864_finite__linorder__max__induct,axiom,
    ! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [B4: extended_enat,A6: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A6 )
             => ( ! [X2: extended_enat] :
                    ( ( member_Extended_enat @ X2 @ A6 )
                   => ( ord_le72135733267957522d_enat @ X2 @ B4 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_Extended_enat @ B4 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4865_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B4: real,A6: set_real] :
              ( ( finite_finite_real @ A6 )
             => ( ! [X2: real] :
                    ( ( member_real @ X2 @ A6 )
                   => ( ord_less_real @ X2 @ B4 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_real @ B4 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4866_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B4: int,A6: set_int] :
              ( ( finite_finite_int @ A6 )
             => ( ! [X2: int] :
                    ( ( member_int @ X2 @ A6 )
                   => ( ord_less_int @ X2 @ B4 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_int @ B4 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4867_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B4: nat,A6: set_nat] :
              ( ( finite_finite_nat @ A6 )
             => ( ! [X2: nat] :
                    ( ( member_nat @ X2 @ A6 )
                   => ( ord_less_nat @ B4 @ X2 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_nat @ B4 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4868_finite__linorder__min__induct,axiom,
    ! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [B4: extended_enat,A6: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A6 )
             => ( ! [X2: extended_enat] :
                    ( ( member_Extended_enat @ X2 @ A6 )
                   => ( ord_le72135733267957522d_enat @ B4 @ X2 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_Extended_enat @ B4 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4869_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B4: real,A6: set_real] :
              ( ( finite_finite_real @ A6 )
             => ( ! [X2: real] :
                    ( ( member_real @ X2 @ A6 )
                   => ( ord_less_real @ B4 @ X2 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_real @ B4 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4870_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B4: int,A6: set_int] :
              ( ( finite_finite_int @ A6 )
             => ( ! [X2: int] :
                    ( ( member_int @ X2 @ A6 )
                   => ( ord_less_int @ B4 @ X2 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_int @ B4 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4871_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A4: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( member_set_nat @ A4 @ A2 )
                 => ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
                   => ( ~ ( member_set_nat @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_set_nat @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4872_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A4: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( member_complex @ A4 @ A2 )
                 => ( ( ord_le211207098394363844omplex @ F4 @ A2 )
                   => ( ~ ( member_complex @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_complex @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4873_finite__subset__induct_H,axiom,
    ! [F3: set_Extended_enat,A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ord_le7203529160286727270d_enat @ F3 @ A2 )
       => ( ( P @ bot_bo7653980558646680370d_enat )
         => ( ! [A4: extended_enat,F4: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F4 )
               => ( ( member_Extended_enat @ A4 @ A2 )
                 => ( ( ord_le7203529160286727270d_enat @ F4 @ A2 )
                   => ( ~ ( member_Extended_enat @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_Extended_enat @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4874_finite__subset__induct_H,axiom,
    ! [F3: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( ord_less_eq_set_real @ F3 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A4: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( member_real @ A4 @ A2 )
                 => ( ( ord_less_eq_set_real @ F4 @ A2 )
                   => ( ~ ( member_real @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_real @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4875_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ord_less_eq_set_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A4: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat @ A4 @ A2 )
                 => ( ( ord_less_eq_set_nat @ F4 @ A2 )
                   => ( ~ ( member_nat @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4876_finite__subset__induct_H,axiom,
    ! [F3: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( ord_less_eq_set_int @ F3 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A4: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( member_int @ A4 @ A2 )
                 => ( ( ord_less_eq_set_int @ F4 @ A2 )
                   => ( ~ ( member_int @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_int @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4877_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A4: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( member_set_nat @ A4 @ A2 )
                 => ( ~ ( member_set_nat @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_set_nat @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4878_finite__subset__induct,axiom,
    ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A4: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( member_complex @ A4 @ A2 )
                 => ( ~ ( member_complex @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_complex @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4879_finite__subset__induct,axiom,
    ! [F3: set_Extended_enat,A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ord_le7203529160286727270d_enat @ F3 @ A2 )
       => ( ( P @ bot_bo7653980558646680370d_enat )
         => ( ! [A4: extended_enat,F4: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F4 )
               => ( ( member_Extended_enat @ A4 @ A2 )
                 => ( ~ ( member_Extended_enat @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_Extended_enat @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4880_finite__subset__induct,axiom,
    ! [F3: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( ord_less_eq_set_real @ F3 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A4: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( member_real @ A4 @ A2 )
                 => ( ~ ( member_real @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_real @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4881_finite__subset__induct,axiom,
    ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ord_less_eq_set_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A4: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat @ A4 @ A2 )
                 => ( ~ ( member_nat @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4882_finite__subset__induct,axiom,
    ! [F3: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( ord_less_eq_set_int @ F3 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A4: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( member_int @ A4 @ A2 )
                 => ( ~ ( member_int @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_int @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4883_finite__remove__induct,axiom,
    ! [B: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [A6: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A6 )
             => ( ( A6 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A6 @ B )
                 => ( ! [X2: set_nat] :
                        ( ( member_set_nat @ X2 @ A6 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_4884_finite__remove__induct,axiom,
    ! [B: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A6: set_complex] :
              ( ( finite3207457112153483333omplex @ A6 )
             => ( ( A6 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A6 @ B )
                 => ( ! [X2: complex] :
                        ( ( member_complex @ X2 @ A6 )
                       => ( P @ ( minus_811609699411566653omplex @ A6 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_4885_finite__remove__induct,axiom,
    ! [B: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [A6: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A6 )
             => ( ( A6 != bot_bo7653980558646680370d_enat )
               => ( ( ord_le7203529160286727270d_enat @ A6 @ B )
                 => ( ! [X2: extended_enat] :
                        ( ( member_Extended_enat @ X2 @ A6 )
                       => ( P @ ( minus_925952699566721837d_enat @ A6 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_4886_finite__remove__induct,axiom,
    ! [B: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A6: set_real] :
              ( ( finite_finite_real @ A6 )
             => ( ( A6 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A6 @ B )
                 => ( ! [X2: real] :
                        ( ( member_real @ X2 @ A6 )
                       => ( P @ ( minus_minus_set_real @ A6 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_4887_finite__remove__induct,axiom,
    ! [B: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A6: set_nat] :
              ( ( finite_finite_nat @ A6 )
             => ( ( A6 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A6 @ B )
                 => ( ! [X2: nat] :
                        ( ( member_nat @ X2 @ A6 )
                       => ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_4888_finite__remove__induct,axiom,
    ! [B: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A6: set_int] :
              ( ( finite_finite_int @ A6 )
             => ( ( A6 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A6 @ B )
                 => ( ! [X2: int] :
                        ( ( member_int @ X2 @ A6 )
                       => ( P @ ( minus_minus_set_int @ A6 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_4889_remove__induct,axiom,
    ! [P: set_set_nat > $o,B: set_set_nat] :
      ( ( P @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B )
         => ( P @ B ) )
       => ( ! [A6: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A6 )
             => ( ( A6 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A6 @ B )
                 => ( ! [X2: set_nat] :
                        ( ( member_set_nat @ X2 @ A6 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A6 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_4890_remove__induct,axiom,
    ! [P: set_complex > $o,B: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B )
         => ( P @ B ) )
       => ( ! [A6: set_complex] :
              ( ( finite3207457112153483333omplex @ A6 )
             => ( ( A6 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A6 @ B )
                 => ( ! [X2: complex] :
                        ( ( member_complex @ X2 @ A6 )
                       => ( P @ ( minus_811609699411566653omplex @ A6 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_4891_remove__induct,axiom,
    ! [P: set_Extended_enat > $o,B: set_Extended_enat] :
      ( ( P @ bot_bo7653980558646680370d_enat )
     => ( ( ~ ( finite4001608067531595151d_enat @ B )
         => ( P @ B ) )
       => ( ! [A6: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A6 )
             => ( ( A6 != bot_bo7653980558646680370d_enat )
               => ( ( ord_le7203529160286727270d_enat @ A6 @ B )
                 => ( ! [X2: extended_enat] :
                        ( ( member_Extended_enat @ X2 @ A6 )
                       => ( P @ ( minus_925952699566721837d_enat @ A6 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_4892_remove__induct,axiom,
    ! [P: set_real > $o,B: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B )
         => ( P @ B ) )
       => ( ! [A6: set_real] :
              ( ( finite_finite_real @ A6 )
             => ( ( A6 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A6 @ B )
                 => ( ! [X2: real] :
                        ( ( member_real @ X2 @ A6 )
                       => ( P @ ( minus_minus_set_real @ A6 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_4893_remove__induct,axiom,
    ! [P: set_nat > $o,B: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B )
         => ( P @ B ) )
       => ( ! [A6: set_nat] :
              ( ( finite_finite_nat @ A6 )
             => ( ( A6 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A6 @ B )
                 => ( ! [X2: nat] :
                        ( ( member_nat @ X2 @ A6 )
                       => ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_4894_remove__induct,axiom,
    ! [P: set_int > $o,B: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B )
         => ( P @ B ) )
       => ( ! [A6: set_int] :
              ( ( finite_finite_int @ A6 )
             => ( ( A6 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A6 @ B )
                 => ( ! [X2: int] :
                        ( ( member_int @ X2 @ A6 )
                       => ( P @ ( minus_minus_set_int @ A6 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
                   => ( P @ A6 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_4895_atLeast0__atMost__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_4896_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_4897_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_4898_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_4899_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_4900_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_4901_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X @ B ) )
      = ( ( ( member_set_nat @ X @ B )
         => ( ord_less_set_set_nat @ A2 @ B ) )
        & ( ~ ( member_set_nat @ X @ B )
         => ( ( ( member_set_nat @ X @ A2 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B ) )
            & ( ~ ( member_set_nat @ X @ A2 )
             => ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4902_psubset__insert__iff,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,B: set_Extended_enat] :
      ( ( ord_le2529575680413868914d_enat @ A2 @ ( insert_Extended_enat @ X @ B ) )
      = ( ( ( member_Extended_enat @ X @ B )
         => ( ord_le2529575680413868914d_enat @ A2 @ B ) )
        & ( ~ ( member_Extended_enat @ X @ B )
         => ( ( ( member_Extended_enat @ X @ A2 )
             => ( ord_le2529575680413868914d_enat @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) @ B ) )
            & ( ~ ( member_Extended_enat @ X @ A2 )
             => ( ord_le7203529160286727270d_enat @ A2 @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4903_psubset__insert__iff,axiom,
    ! [A2: set_real,X: real,B: set_real] :
      ( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B ) )
      = ( ( ( member_real @ X @ B )
         => ( ord_less_set_real @ A2 @ B ) )
        & ( ~ ( member_real @ X @ B )
         => ( ( ( member_real @ X @ A2 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B ) )
            & ( ~ ( member_real @ X @ A2 )
             => ( ord_less_eq_set_real @ A2 @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4904_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B ) )
      = ( ( ( member_nat @ X @ B )
         => ( ord_less_set_nat @ A2 @ B ) )
        & ( ~ ( member_nat @ X @ B )
         => ( ( ( member_nat @ X @ A2 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
            & ( ~ ( member_nat @ X @ A2 )
             => ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4905_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B: set_int] :
      ( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B ) )
      = ( ( ( member_int @ X @ B )
         => ( ord_less_set_int @ A2 @ B ) )
        & ( ~ ( member_int @ X @ B )
         => ( ( ( member_int @ X @ A2 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B ) )
            & ( ~ ( member_int @ X @ A2 )
             => ( ord_less_eq_set_int @ A2 @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4906_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) )
        = ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_4907_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) )
        = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_4908_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( set_or1269000886237332187st_nat @ M2 @ N2 )
        = ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_4909_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_4910_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_4911_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_4912_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_4913_set__replicate__Suc,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N2 ) @ X ) )
      = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_4914_set__replicate__Suc,axiom,
    ! [N2: nat,X: extended_enat] :
      ( ( set_Extended_enat2 @ ( replic7216382294607269926d_enat @ ( suc @ N2 ) @ X ) )
      = ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ).

% set_replicate_Suc
thf(fact_4915_set__replicate__Suc,axiom,
    ! [N2: nat,X: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N2 ) @ X ) )
      = ( insert_real @ X @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_4916_set__replicate__Suc,axiom,
    ! [N2: nat,X: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N2 ) @ X ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_4917_set__replicate__Suc,axiom,
    ! [N2: nat,X: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N2 ) @ X ) )
      = ( insert_int @ X @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_4918_set__replicate__conv__if,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X ) )
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4919_set__replicate__conv__if,axiom,
    ! [N2: nat,X: extended_enat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_Extended_enat2 @ ( replic7216382294607269926d_enat @ N2 @ X ) )
          = bot_bo7653980558646680370d_enat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_Extended_enat2 @ ( replic7216382294607269926d_enat @ N2 @ X ) )
          = ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4920_set__replicate__conv__if,axiom,
    ! [N2: nat,X: real] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N2 @ X ) )
          = bot_bot_set_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N2 @ X ) )
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4921_set__replicate__conv__if,axiom,
    ! [N2: nat,X: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
          = bot_bot_set_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4922_set__replicate__conv__if,axiom,
    ! [N2: nat,X: int] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N2 @ X ) )
          = bot_bot_set_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N2 @ X ) )
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4923_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z3: nat] :
      ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z3 ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z3 ) )
        = ( insert_nat @ N2 @ ( nat_set_decode @ Z3 ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_4924_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_4925_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_4926_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M: nat,N: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N = zero_zero_nat )
            | ( ord_less_nat @ M @ N ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% divmod_nat_if
thf(fact_4927_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_4928_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_4929_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K2: int] :
          ( if_complex @ ( K2 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_4930_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K2: int] :
          ( if_int @ ( K2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_4931_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K2: int] :
          ( if_real @ ( K2 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_4932_concat__bit__Suc,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_4933_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4934_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4935_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4936_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4937_of__int__eq__iff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ( ring_1_of_int_real @ W2 )
        = ( ring_1_of_int_real @ Z3 ) )
      = ( W2 = Z3 ) ) ).

% of_int_eq_iff
thf(fact_4938_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% mask_nat_positive_iff
thf(fact_4939_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_int_of_bool
thf(fact_4940_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_int_of_bool
thf(fact_4941_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_4942_of__int__eq__0__iff,axiom,
    ! [Z3: int] :
      ( ( ( ring_1_of_int_int @ Z3 )
        = zero_zero_int )
      = ( Z3 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4943_of__int__eq__0__iff,axiom,
    ! [Z3: int] :
      ( ( ( ring_17405671764205052669omplex @ Z3 )
        = zero_zero_complex )
      = ( Z3 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4944_of__int__eq__0__iff,axiom,
    ! [Z3: int] :
      ( ( ( ring_1_of_int_real @ Z3 )
        = zero_zero_real )
      = ( Z3 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4945_of__int__0__eq__iff,axiom,
    ! [Z3: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z3 ) )
      = ( Z3 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4946_of__int__0__eq__iff,axiom,
    ! [Z3: int] :
      ( ( zero_zero_complex
        = ( ring_17405671764205052669omplex @ Z3 ) )
      = ( Z3 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4947_of__int__0__eq__iff,axiom,
    ! [Z3: int] :
      ( ( zero_zero_real
        = ( ring_1_of_int_real @ Z3 ) )
      = ( Z3 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4948_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_4949_of__int__0,axiom,
    ( ( ring_17405671764205052669omplex @ zero_zero_int )
    = zero_zero_complex ) ).

% of_int_0
thf(fact_4950_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_4951_of__int__le__iff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) )
      = ( ord_less_eq_int @ W2 @ Z3 ) ) ).

% of_int_le_iff
thf(fact_4952_of__int__le__iff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_eq_int @ W2 @ Z3 ) ) ).

% of_int_le_iff
thf(fact_4953_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_4954_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_4955_of__int__eq__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ( ring_1_of_int_int @ Z3 )
        = ( numeral_numeral_int @ N2 ) )
      = ( Z3
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4956_of__int__eq__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ( ring_1_of_int_real @ Z3 )
        = ( numeral_numeral_real @ N2 ) )
      = ( Z3
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4957_of__int__less__iff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) )
      = ( ord_less_int @ W2 @ Z3 ) ) ).

% of_int_less_iff
thf(fact_4958_of__int__less__iff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_int @ W2 @ Z3 ) ) ).

% of_int_less_iff
thf(fact_4959_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_4960_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_4961_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_4962_of__int__eq__1__iff,axiom,
    ! [Z3: int] :
      ( ( ( ring_1_of_int_int @ Z3 )
        = one_one_int )
      = ( Z3 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4963_of__int__eq__1__iff,axiom,
    ! [Z3: int] :
      ( ( ( ring_17405671764205052669omplex @ Z3 )
        = one_one_complex )
      = ( Z3 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4964_of__int__eq__1__iff,axiom,
    ! [Z3: int] :
      ( ( ( ring_1_of_int_real @ Z3 )
        = one_one_real )
      = ( Z3 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4965_of__int__mult,axiom,
    ! [W2: int,Z3: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z3 ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) ) ) ).

% of_int_mult
thf(fact_4966_of__int__mult,axiom,
    ! [W2: int,Z3: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z3 ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) ) ) ).

% of_int_mult
thf(fact_4967_of__int__mult,axiom,
    ! [W2: int,Z3: int] :
      ( ( ring_17405671764205052669omplex @ ( times_times_int @ W2 @ Z3 ) )
      = ( times_times_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z3 ) ) ) ).

% of_int_mult
thf(fact_4968_of__int__add,axiom,
    ! [W2: int,Z3: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z3 ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) ) ) ).

% of_int_add
thf(fact_4969_of__int__add,axiom,
    ! [W2: int,Z3: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W2 @ Z3 ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) ) ) ).

% of_int_add
thf(fact_4970_of__int__minus,axiom,
    ! [Z3: int] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z3 ) )
      = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).

% of_int_minus
thf(fact_4971_of__int__minus,axiom,
    ! [Z3: int] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z3 ) )
      = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z3 ) ) ) ).

% of_int_minus
thf(fact_4972_mask__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
    = zero_zero_int ) ).

% mask_0
thf(fact_4973_mask__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% mask_0
thf(fact_4974_mask__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2000444600071755411sk_int @ N2 )
        = zero_zero_int )
      = ( N2 = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_4975_mask__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2002935070580805687sk_nat @ N2 )
        = zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_4976_of__int__diff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z3 ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) ) ) ).

% of_int_diff
thf(fact_4977_of__int__diff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W2 @ Z3 ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) ) ) ).

% of_int_diff
thf(fact_4978_of__int__power,axiom,
    ! [Z3: int,N2: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z3 @ N2 ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z3 ) @ N2 ) ) ).

% of_int_power
thf(fact_4979_of__int__power,axiom,
    ! [Z3: int,N2: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z3 @ N2 ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z3 ) @ N2 ) ) ).

% of_int_power
thf(fact_4980_of__int__power,axiom,
    ! [Z3: int,N2: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z3 @ N2 ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z3 ) @ N2 ) ) ).

% of_int_power
thf(fact_4981_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W2 )
        = ( ring_1_of_int_real @ X ) )
      = ( ( power_power_int @ B2 @ W2 )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4982_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B2 ) @ W2 )
        = ( ring_17405671764205052669omplex @ X ) )
      = ( ( power_power_int @ B2 @ W2 )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4983_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W2 )
        = ( ring_1_of_int_int @ X ) )
      = ( ( power_power_int @ B2 @ W2 )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4984_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( ( ( ring_1_of_int_real @ X )
        = ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W2 ) )
      = ( X
        = ( power_power_int @ B2 @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4985_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( ( ( ring_17405671764205052669omplex @ X )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B2 ) @ W2 ) )
      = ( X
        = ( power_power_int @ B2 @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4986_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( ( ( ring_1_of_int_int @ X )
        = ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W2 ) )
      = ( X
        = ( power_power_int @ B2 @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4987_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_4988_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_4989_of__int__0__le__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).

% of_int_0_le_iff
thf(fact_4990_of__int__0__le__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).

% of_int_0_le_iff
thf(fact_4991_of__int__le__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4992_of__int__le__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_4993_of__int__0__less__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) )
      = ( ord_less_int @ zero_zero_int @ Z3 ) ) ).

% of_int_0_less_iff
thf(fact_4994_of__int__0__less__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_int @ zero_zero_int @ Z3 ) ) ).

% of_int_0_less_iff
thf(fact_4995_of__int__less__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ zero_zero_real )
      = ( ord_less_int @ Z3 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4996_of__int__less__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
      = ( ord_less_int @ Z3 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_4997_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z3: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z3 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z3 ) ) ).

% of_int_numeral_le_iff
thf(fact_4998_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z3: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z3 ) ) ).

% of_int_numeral_le_iff
thf(fact_4999_of__int__le__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_int @ Z3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5000_of__int__le__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_int @ Z3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5001_of__int__less__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_int @ Z3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5002_of__int__less__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_int @ Z3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5003_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z3: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z3 ) ) ).

% of_int_numeral_less_iff
thf(fact_5004_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z3: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z3 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z3 ) ) ).

% of_int_numeral_less_iff
thf(fact_5005_of__int__1__le__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z3 ) )
      = ( ord_less_eq_int @ one_one_int @ Z3 ) ) ).

% of_int_1_le_iff
thf(fact_5006_of__int__1__le__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_eq_int @ one_one_int @ Z3 ) ) ).

% of_int_1_le_iff
thf(fact_5007_of__int__le__1__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ one_one_real )
      = ( ord_less_eq_int @ Z3 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5008_of__int__le__1__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ one_one_int )
      = ( ord_less_eq_int @ Z3 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5009_of__int__less__1__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ one_one_real )
      = ( ord_less_int @ Z3 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5010_of__int__less__1__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ one_one_int )
      = ( ord_less_int @ Z3 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5011_of__int__1__less__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z3 ) )
      = ( ord_less_int @ one_one_int @ Z3 ) ) ).

% of_int_1_less_iff
thf(fact_5012_of__int__1__less__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_int @ one_one_int @ Z3 ) ) ).

% of_int_1_less_iff
thf(fact_5013_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W2 ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B2 @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5014_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W2 ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B2 @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5015_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B2 @ W2 ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5016_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B2 @ W2 ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5017_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5018_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5019_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5020_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5021_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5022_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5023_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5024_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5025_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5026_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_5027_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_5028_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_5029_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( power_power_int @ B2 @ W2 ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5030_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W2: nat,X: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( power_power_int @ B2 @ W2 ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5031_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W2 ) )
      = ( ord_less_int @ X @ ( power_power_int @ B2 @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5032_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W2 ) )
      = ( ord_less_int @ X @ ( power_power_int @ B2 @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5033_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5034_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5035_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5036_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5037_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5038_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5039_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5040_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5041_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5042_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5043_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5044_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5045_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5046_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5047_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5048_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5049_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5050_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5051_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ).

% ex_le_of_int
thf(fact_5052_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ).

% ex_of_int_less
thf(fact_5053_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ).

% ex_less_of_int
thf(fact_5054_mult__of__int__commute,axiom,
    ! [X: int,Y: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5055_mult__of__int__commute,axiom,
    ! [X: int,Y: real] :
      ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5056_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] :
      ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( ring_17405671764205052669omplex @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5057_of__int__max,axiom,
    ! [X: int,Y: int] :
      ( ( ring_1_of_int_real @ ( ord_max_int @ X @ Y ) )
      = ( ord_max_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).

% of_int_max
thf(fact_5058_of__int__max,axiom,
    ! [X: int,Y: int] :
      ( ( ring_1_of_int_int @ ( ord_max_int @ X @ Y ) )
      = ( ord_max_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).

% of_int_max
thf(fact_5059_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_5060_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M2: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M2 @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M2 @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_5061_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_eq_int @ M2 @ ( plus_plus_int @ one_one_int @ N2 ) )
     => ( ( set_or1266510415728281911st_int @ M2 @ ( plus_plus_int @ one_one_int @ N2 ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M2 @ N2 ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_5062_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J2: int] : ( if_set_int @ ( ord_less_int @ J2 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) ) ) ) ).

% simp_from_to
thf(fact_5063_less__mask,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).

% less_mask
thf(fact_5064_of__int__nonneg,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) ) ) ).

% of_int_nonneg
thf(fact_5065_of__int__nonneg,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).

% of_int_nonneg
thf(fact_5066_of__int__pos,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) ) ) ).

% of_int_pos
thf(fact_5067_of__int__pos,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).

% of_int_pos
thf(fact_5068_floor__exists1,axiom,
    ! [X: real] :
    ? [X5: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X5 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y6: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y6 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
         => ( Y6 = X5 ) ) ) ).

% floor_exists1
thf(fact_5069_floor__exists,axiom,
    ! [X: real] :
    ? [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5070_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera6690914467698888265omplex @ ( inc @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% numeral_inc
thf(fact_5071_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_nat @ ( inc @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_5072_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera1916890842035813515d_enat @ ( inc @ X ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_inc
thf(fact_5073_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_int @ ( inc @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_5074_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_real @ ( inc @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% numeral_inc
thf(fact_5075_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_5076_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_5077_Suc__mask__eq__exp,axiom,
    ! [N2: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_mask_eq_exp
thf(fact_5078_mask__nat__less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% mask_nat_less_exp
thf(fact_5079_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_5080_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_5081_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_5082_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_5083_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_5084_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_5085_Sum__Icc__int,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_eq_int @ M2 @ N2 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X4: int] : X4
          @ ( set_or1266510415728281911st_int @ M2 @ N2 ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M2 @ ( minus_minus_int @ M2 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_5086_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_5087_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L2: int] :
          ( if_int
          @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_5088_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_5089_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_5090_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_5091_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_5092_bit_Oconj__zero__left,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_5093_bit_Oconj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_5094_sum_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A2 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_5095_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A2 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_5096_sum_Oneutral__const,axiom,
    ! [A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A2 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_5097_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A2 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_5098_sum_Oempty,axiom,
    ! [G: extended_enat > nat] :
      ( ( groups2027974829824023292at_nat @ G @ bot_bo7653980558646680370d_enat )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_5099_sum_Oempty,axiom,
    ! [G: extended_enat > real] :
      ( ( groups4148127829035722712t_real @ G @ bot_bo7653980558646680370d_enat )
      = zero_zero_real ) ).

% sum.empty
thf(fact_5100_sum_Oempty,axiom,
    ! [G: extended_enat > int] :
      ( ( groups2025484359314973016at_int @ G @ bot_bo7653980558646680370d_enat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_5101_sum_Oempty,axiom,
    ! [G: extended_enat > complex] :
      ( ( groups6818542070133387226omplex @ G @ bot_bo7653980558646680370d_enat )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_5102_sum_Oempty,axiom,
    ! [G: extended_enat > extended_enat] :
      ( ( groups2433450451889696826d_enat @ G @ bot_bo7653980558646680370d_enat )
      = zero_z5237406670263579293d_enat ) ).

% sum.empty
thf(fact_5103_sum_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_5104_sum_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_5105_sum_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty
thf(fact_5106_sum_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_5107_sum_Oempty,axiom,
    ! [G: real > extended_enat] :
      ( ( groups2800946370649118462d_enat @ G @ bot_bot_set_real )
      = zero_z5237406670263579293d_enat ) ).

% sum.empty
thf(fact_5108_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_5109_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_5110_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2027974829824023292at_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_5111_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_5112_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_5113_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > real] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_5114_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups3539618377306564664at_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_5115_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_5116_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > int] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2025484359314973016at_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_5117_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_5118_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ F3 )
             => ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5119_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: int > nat] :
      ( ( finite_finite_int @ F3 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ F3 )
             => ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5120_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ( groups2027974829824023292at_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ F3 )
             => ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5121_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ( groups7108830773950497114d_enat @ F @ F3 )
          = zero_z5237406670263579293d_enat )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ F3 )
             => ( ( F @ X4 )
                = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5122_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ( groups1752964319039525884d_enat @ F @ F3 )
          = zero_z5237406670263579293d_enat )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ F3 )
             => ( ( F @ X4 )
                = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5123_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: int > extended_enat] :
      ( ( finite_finite_int @ F3 )
     => ( ( ( groups4225252721152677374d_enat @ F @ F3 )
          = zero_z5237406670263579293d_enat )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ F3 )
             => ( ( F @ X4 )
                = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5124_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ( groups2433450451889696826d_enat @ F @ F3 )
          = zero_z5237406670263579293d_enat )
        = ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ F3 )
             => ( ( F @ X4 )
                = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5125_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ F3 )
             => ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5126_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_5127_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B2: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_5128_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B2: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_5129_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B2: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_5130_sum_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_5131_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B2: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_5132_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B2: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_5133_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B2: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_5134_sum_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_5135_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B2: real > int] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_int )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_int )
              @ S2 )
            = zero_zero_int ) ) ) ) ).

% sum.delta
thf(fact_5136_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B2: nat > int] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups3539618377306564664at_int
              @ ^ [K2: nat] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_int )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups3539618377306564664at_int
              @ ^ [K2: nat] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ zero_zero_int )
              @ S2 )
            = zero_zero_int ) ) ) ) ).

% sum.delta
thf(fact_5137_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_5138_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B2: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_5139_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B2: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K2: int] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K2: int] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_5140_sum_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_nat )
              @ S2 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_5141_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_5142_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B2: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_5143_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B2: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_5144_sum_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_5145_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: real > int] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K2: real] : ( if_int @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_int )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K2: real] : ( if_int @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_int )
              @ S2 )
            = zero_zero_int ) ) ) ) ).

% sum.delta'
thf(fact_5146_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B2: nat > int] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups3539618377306564664at_int
              @ ^ [K2: nat] : ( if_int @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_int )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups3539618377306564664at_int
              @ ^ [K2: nat] : ( if_int @ ( A = K2 ) @ ( B2 @ K2 ) @ zero_zero_int )
              @ S2 )
            = zero_zero_int ) ) ) ) ).

% sum.delta'
thf(fact_5147_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5148_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5149_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5150_sum_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ~ ( member_Extended_enat @ X @ A2 )
       => ( ( groups2027974829824023292at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5151_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5152_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5153_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5154_sum_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ~ ( member_Extended_enat @ X @ A2 )
       => ( ( groups2025484359314973016at_int @ G @ ( insert_Extended_enat @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups2025484359314973016at_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5155_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5156_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5157_of__int__sum,axiom,
    ! [F: complex > int,A2: set_complex] :
      ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X4: complex] : ( ring_17405671764205052669omplex @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_5158_of__int__sum,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X4: nat] : ( ring_1_of_int_real @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_5159_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups8778361861064173332t_real
        @ ^ [X4: int] : ( ring_1_of_int_real @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_5160_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( ring_1_of_int_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_5161_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_5162_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_5163_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_5164_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_5165_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_5166_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_5167_sum_Oneutral,axiom,
    ! [A2: set_int,G: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_5168_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_5169_sum_Oneutral,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_5170_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_5171_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: extended_enat > nat,A2: set_Extended_enat] :
      ( ( ( groups2027974829824023292at_nat @ G @ A2 )
       != zero_zero_nat )
     => ~ ! [A4: extended_enat] :
            ( ( member_Extended_enat @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5172_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > nat,A2: set_real] :
      ( ( ( groups1935376822645274424al_nat @ G @ A2 )
       != zero_zero_nat )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5173_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > nat,A2: set_int] :
      ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
       != zero_zero_nat )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5174_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: extended_enat > real,A2: set_Extended_enat] :
      ( ( ( groups4148127829035722712t_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A4: extended_enat] :
            ( ( member_Extended_enat @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5175_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A2: set_real] :
      ( ( ( groups8097168146408367636l_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5176_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A2: set_int] :
      ( ( ( groups8778361861064173332t_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5177_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: extended_enat > int,A2: set_Extended_enat] :
      ( ( ( groups2025484359314973016at_int @ G @ A2 )
       != zero_zero_int )
     => ~ ! [A4: extended_enat] :
            ( ( member_Extended_enat @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5178_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > int,A2: set_real] :
      ( ( ( groups1932886352136224148al_int @ G @ A2 )
       != zero_zero_int )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5179_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > int,A2: set_nat] :
      ( ( ( groups3539618377306564664at_int @ G @ A2 )
       != zero_zero_int )
     => ~ ! [A4: nat] :
            ( ( member_nat @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5180_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: extended_enat > complex,A2: set_Extended_enat] :
      ( ( ( groups6818542070133387226omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A4: extended_enat] :
            ( ( member_Extended_enat @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5181_sum__mono,axiom,
    ! [K5: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
      ( ! [I4: extended_enat] :
          ( ( member_Extended_enat @ I4 @ K5 )
         => ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ K5 ) @ ( groups4148127829035722712t_real @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5182_sum__mono,axiom,
    ! [K5: set_real,F: real > real,G: real > real] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ K5 )
         => ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K5 ) @ ( groups8097168146408367636l_real @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5183_sum__mono,axiom,
    ! [K5: set_int,F: int > real,G: int > real] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ K5 )
         => ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ K5 ) @ ( groups8778361861064173332t_real @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5184_sum__mono,axiom,
    ! [K5: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ! [I4: extended_enat] :
          ( ( member_Extended_enat @ I4 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_nat @ ( groups2027974829824023292at_nat @ F @ K5 ) @ ( groups2027974829824023292at_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5185_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5186_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5187_sum__mono,axiom,
    ! [K5: set_Extended_enat,F: extended_enat > int,G: extended_enat > int] :
      ( ! [I4: extended_enat] :
          ( ( member_Extended_enat @ I4 @ K5 )
         => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_int @ ( groups2025484359314973016at_int @ F @ K5 ) @ ( groups2025484359314973016at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5188_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ K5 )
         => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5189_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ K5 )
         => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5190_sum__mono,axiom,
    ! [K5: set_int,F: int > int,G: int > int] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ K5 )
         => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K5 ) @ ( groups4538972089207619220nt_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5191_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( plus_plus_int @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A2 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5192_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( plus_plus_nat @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5193_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X4: complex] : ( plus_plus_complex @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5194_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X4: nat] : ( plus_plus_real @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5195_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_int,G: real > int > int,R: real > int > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_int @ B )
       => ( ( groups1932886352136224148al_int
            @ ^ [X4: real] :
                ( groups4538972089207619220nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y5: int] :
                      ( ( member_int @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y5: int] :
                ( groups1932886352136224148al_int
                @ ^ [X4: real] : ( G @ X4 @ Y5 )
                @ ( collect_real
                  @ ^ [X4: real] :
                      ( ( member_real @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% sum.swap_restrict
thf(fact_5196_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B: set_int,G: nat > int > int,R: nat > int > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_int @ B )
       => ( ( groups3539618377306564664at_int
            @ ^ [X4: nat] :
                ( groups4538972089207619220nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y5: int] :
                      ( ( member_int @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y5: int] :
                ( groups3539618377306564664at_int
                @ ^ [X4: nat] : ( G @ X4 @ Y5 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% sum.swap_restrict
thf(fact_5197_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B: set_int,G: complex > int > int,R: complex > int > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_int @ B )
       => ( ( groups5690904116761175830ex_int
            @ ^ [X4: complex] :
                ( groups4538972089207619220nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y5: int] :
                      ( ( member_int @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y5: int] :
                ( groups5690904116761175830ex_int
                @ ^ [X4: complex] : ( G @ X4 @ Y5 )
                @ ( collect_complex
                  @ ^ [X4: complex] :
                      ( ( member_complex @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% sum.swap_restrict
thf(fact_5198_sum_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B: set_int,G: extended_enat > int > int,R: extended_enat > int > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite_finite_int @ B )
       => ( ( groups2025484359314973016at_int
            @ ^ [X4: extended_enat] :
                ( groups4538972089207619220nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y5: int] :
                      ( ( member_int @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y5: int] :
                ( groups2025484359314973016at_int
                @ ^ [X4: extended_enat] : ( G @ X4 @ Y5 )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X4: extended_enat] :
                      ( ( member_Extended_enat @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% sum.swap_restrict
thf(fact_5199_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_nat,G: real > nat > nat,R: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups1935376822645274424al_nat
            @ ^ [X4: real] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y5: nat] :
                ( groups1935376822645274424al_nat
                @ ^ [X4: real] : ( G @ X4 @ Y5 )
                @ ( collect_real
                  @ ^ [X4: real] :
                      ( ( member_real @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% sum.swap_restrict
thf(fact_5200_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups5693394587270226106ex_nat
            @ ^ [X4: complex] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y5: nat] :
                ( groups5693394587270226106ex_nat
                @ ^ [X4: complex] : ( G @ X4 @ Y5 )
                @ ( collect_complex
                  @ ^ [X4: complex] :
                      ( ( member_complex @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% sum.swap_restrict
thf(fact_5201_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B: set_nat,G: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X4: int] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y5: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X4: int] : ( G @ X4 @ Y5 )
                @ ( collect_int
                  @ ^ [X4: int] :
                      ( ( member_int @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% sum.swap_restrict
thf(fact_5202_sum_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B: set_nat,G: extended_enat > nat > nat,R: extended_enat > nat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups2027974829824023292at_nat
            @ ^ [X4: extended_enat] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y5: nat] :
                ( groups2027974829824023292at_nat
                @ ^ [X4: extended_enat] : ( G @ X4 @ Y5 )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X4: extended_enat] :
                      ( ( member_Extended_enat @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% sum.swap_restrict
thf(fact_5203_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_complex,G: real > complex > complex,R: real > complex > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite3207457112153483333omplex @ B )
       => ( ( groups5754745047067104278omplex
            @ ^ [X4: real] :
                ( groups7754918857620584856omplex @ ( G @ X4 )
                @ ( collect_complex
                  @ ^ [Y5: complex] :
                      ( ( member_complex @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y5: complex] :
                ( groups5754745047067104278omplex
                @ ^ [X4: real] : ( G @ X4 @ Y5 )
                @ ( collect_real
                  @ ^ [X4: real] :
                      ( ( member_real @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% sum.swap_restrict
thf(fact_5204_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite3207457112153483333omplex @ B )
       => ( ( groups2073611262835488442omplex
            @ ^ [X4: nat] :
                ( groups7754918857620584856omplex @ ( G @ X4 )
                @ ( collect_complex
                  @ ^ [Y5: complex] :
                      ( ( member_complex @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y5: complex] :
                ( groups2073611262835488442omplex
                @ ^ [X4: nat] : ( G @ X4 @ Y5 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% sum.swap_restrict
thf(fact_5205_sum__nonpos,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ zero_z5237406670263579293d_enat ) )
     => ( ord_le2932123472753598470d_enat @ ( groups2433450451889696826d_enat @ F @ A2 ) @ zero_z5237406670263579293d_enat ) ) ).

% sum_nonpos
thf(fact_5206_sum__nonpos,axiom,
    ! [A2: set_real,F: real > extended_enat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ zero_z5237406670263579293d_enat ) )
     => ( ord_le2932123472753598470d_enat @ ( groups2800946370649118462d_enat @ F @ A2 ) @ zero_z5237406670263579293d_enat ) ) ).

% sum_nonpos
thf(fact_5207_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > extended_enat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ zero_z5237406670263579293d_enat ) )
     => ( ord_le2932123472753598470d_enat @ ( groups7108830773950497114d_enat @ F @ A2 ) @ zero_z5237406670263579293d_enat ) ) ).

% sum_nonpos
thf(fact_5208_sum__nonpos,axiom,
    ! [A2: set_int,F: int > extended_enat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ zero_z5237406670263579293d_enat ) )
     => ( ord_le2932123472753598470d_enat @ ( groups4225252721152677374d_enat @ F @ A2 ) @ zero_z5237406670263579293d_enat ) ) ).

% sum_nonpos
thf(fact_5209_sum__nonpos,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_5210_sum__nonpos,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_5211_sum__nonpos,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_5212_sum__nonpos,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_5213_sum__nonpos,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_5214_sum__nonpos,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_5215_sum__nonneg,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
     => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( groups2433450451889696826d_enat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5216_sum__nonneg,axiom,
    ! [A2: set_real,F: real > extended_enat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
     => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( groups2800946370649118462d_enat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5217_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > extended_enat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
     => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( groups7108830773950497114d_enat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5218_sum__nonneg,axiom,
    ! [A2: set_int,F: int > extended_enat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
     => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( groups4225252721152677374d_enat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5219_sum__nonneg,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5220_sum__nonneg,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5221_sum__nonneg,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5222_sum__nonneg,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5223_sum__nonneg,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5224_sum__nonneg,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5225_sum__mono__inv,axiom,
    ! [F: real > real,I6: set_real,G: real > real,I: real] :
      ( ( ( groups8097168146408367636l_real @ F @ I6 )
        = ( groups8097168146408367636l_real @ G @ I6 ) )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ I6 )
           => ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_real @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5226_sum__mono__inv,axiom,
    ! [F: complex > real,I6: set_complex,G: complex > real,I: complex] :
      ( ( ( groups5808333547571424918x_real @ F @ I6 )
        = ( groups5808333547571424918x_real @ G @ I6 ) )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ I6 )
           => ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_complex @ I @ I6 )
         => ( ( finite3207457112153483333omplex @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5227_sum__mono__inv,axiom,
    ! [F: int > real,I6: set_int,G: int > real,I: int] :
      ( ( ( groups8778361861064173332t_real @ F @ I6 )
        = ( groups8778361861064173332t_real @ G @ I6 ) )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ I6 )
           => ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_int @ I @ I6 )
         => ( ( finite_finite_int @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5228_sum__mono__inv,axiom,
    ! [F: extended_enat > real,I6: set_Extended_enat,G: extended_enat > real,I: extended_enat] :
      ( ( ( groups4148127829035722712t_real @ F @ I6 )
        = ( groups4148127829035722712t_real @ G @ I6 ) )
     => ( ! [I4: extended_enat] :
            ( ( member_Extended_enat @ I4 @ I6 )
           => ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_Extended_enat @ I @ I6 )
         => ( ( finite4001608067531595151d_enat @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5229_sum__mono__inv,axiom,
    ! [F: real > nat,I6: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I6 )
        = ( groups1935376822645274424al_nat @ G @ I6 ) )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_real @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5230_sum__mono__inv,axiom,
    ! [F: complex > nat,I6: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I6 )
        = ( groups5693394587270226106ex_nat @ G @ I6 ) )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_complex @ I @ I6 )
         => ( ( finite3207457112153483333omplex @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5231_sum__mono__inv,axiom,
    ! [F: int > nat,I6: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I6 )
        = ( groups4541462559716669496nt_nat @ G @ I6 ) )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_int @ I @ I6 )
         => ( ( finite_finite_int @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5232_sum__mono__inv,axiom,
    ! [F: extended_enat > nat,I6: set_Extended_enat,G: extended_enat > nat,I: extended_enat] :
      ( ( ( groups2027974829824023292at_nat @ F @ I6 )
        = ( groups2027974829824023292at_nat @ G @ I6 ) )
     => ( ! [I4: extended_enat] :
            ( ( member_Extended_enat @ I4 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_Extended_enat @ I @ I6 )
         => ( ( finite4001608067531595151d_enat @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5233_sum__mono__inv,axiom,
    ! [F: real > int,I6: set_real,G: real > int,I: real] :
      ( ( ( groups1932886352136224148al_int @ F @ I6 )
        = ( groups1932886352136224148al_int @ G @ I6 ) )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ I6 )
           => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_real @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5234_sum__mono__inv,axiom,
    ! [F: nat > int,I6: set_nat,G: nat > int,I: nat] :
      ( ( ( groups3539618377306564664at_int @ F @ I6 )
        = ( groups3539618377306564664at_int @ G @ I6 ) )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ I6 )
           => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_nat @ I @ I6 )
         => ( ( finite_finite_nat @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5235_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > nat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X4: real] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5236_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > nat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups5693394587270226106ex_nat
          @ ^ [X4: complex] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5237_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > nat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5238_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2027974829824023292at_nat @ G
          @ ( collec4429806609662206161d_enat
            @ ^ [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups2027974829824023292at_nat
          @ ^ [X4: extended_enat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5239_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X4: real] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5240_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X4: complex] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5241_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X4: int] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5242_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > real,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G
          @ ( collec4429806609662206161d_enat
            @ ^ [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups4148127829035722712t_real
          @ ^ [X4: extended_enat] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5243_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > int,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1932886352136224148al_int @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups1932886352136224148al_int
          @ ^ [X4: real] : ( if_int @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_int )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5244_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > int,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups3539618377306564664at_int @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups3539618377306564664at_int
          @ ^ [X4: nat] : ( if_int @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_int )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5245_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > extended_enat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups2800946370649118462d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A2 )
               => ( ( F @ X4 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5246_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups7108830773950497114d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A2 )
               => ( ( F @ X4 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5247_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups1752964319039525884d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A2 )
               => ( ( F @ X4 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5248_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > extended_enat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups4225252721152677374d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A2 )
               => ( ( F @ X4 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5249_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups2433450451889696826d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ A2 )
               => ( ( F @ X4 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5250_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A2 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5251_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A2 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5252_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A2 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5253_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ A2 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5254_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A2 )
               => ( ( F @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5255_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > extended_enat,I: nat > nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups7108830773950497114d_enat @ F @ S ) @ ( groups7108830773950497114d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5256_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: complex > extended_enat,I: complex > nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups7108830773950497114d_enat @ F @ S ) @ ( groups1752964319039525884d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5257_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: int > extended_enat,I: int > nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups7108830773950497114d_enat @ F @ S ) @ ( groups4225252721152677374d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5258_sum__le__included,axiom,
    ! [S: set_nat,T: set_Extended_enat,G: extended_enat > extended_enat,I: extended_enat > nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups7108830773950497114d_enat @ F @ S ) @ ( groups2433450451889696826d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5259_sum__le__included,axiom,
    ! [S: set_complex,T: set_nat,G: nat > extended_enat,I: nat > complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups1752964319039525884d_enat @ F @ S ) @ ( groups7108830773950497114d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5260_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > extended_enat,I: complex > complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups1752964319039525884d_enat @ F @ S ) @ ( groups1752964319039525884d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5261_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > extended_enat,I: int > complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups1752964319039525884d_enat @ F @ S ) @ ( groups4225252721152677374d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5262_sum__le__included,axiom,
    ! [S: set_complex,T: set_Extended_enat,G: extended_enat > extended_enat,I: extended_enat > complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups1752964319039525884d_enat @ F @ S ) @ ( groups2433450451889696826d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5263_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: nat > extended_enat,I: nat > int,F: int > extended_enat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups4225252721152677374d_enat @ F @ S ) @ ( groups7108830773950497114d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5264_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > extended_enat,I: complex > int,F: int > extended_enat] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups4225252721152677374d_enat @ F @ S ) @ ( groups1752964319039525884d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5265_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5266_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5267_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( groups4148127829035722712t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5268_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5269_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5270_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5271_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5272_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5273_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > int,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_int @ ( groups2025484359314973016at_int @ F @ A2 ) @ ( groups2025484359314973016at_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5274_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5275_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5276_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5277_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S2: set_Extended_enat,H2: extended_enat > nat,G: extended_enat > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2027974829824023292at_nat @ H2 @ S2 ) @ ( groups2027974829824023292at_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5278_sum_Orelated,axiom,
    ! [R: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5279_sum_Orelated,axiom,
    ! [R: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5280_sum_Orelated,axiom,
    ! [R: real > real > $o,S2: set_Extended_enat,H2: extended_enat > real,G: extended_enat > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups4148127829035722712t_real @ H2 @ S2 ) @ ( groups4148127829035722712t_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5281_sum_Orelated,axiom,
    ! [R: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5282_sum_Orelated,axiom,
    ! [R: int > int > $o,S2: set_complex,H2: complex > int,G: complex > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5690904116761175830ex_int @ H2 @ S2 ) @ ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5283_sum_Orelated,axiom,
    ! [R: int > int > $o,S2: set_Extended_enat,H2: extended_enat > int,G: extended_enat > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y22 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2025484359314973016at_int @ H2 @ S2 ) @ ( groups2025484359314973016at_int @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5284_sum_Orelated,axiom,
    ! [R: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_complex @ X1 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5285_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5286_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ A2 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5287_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A2 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5288_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A2 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5289_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > extended_enat,G: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_le72135733267957522d_enat @ ( groups1752964319039525884d_enat @ F @ A2 ) @ ( groups1752964319039525884d_enat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5290_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > extended_enat,G: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ A2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_le72135733267957522d_enat @ ( groups2433450451889696826d_enat @ F @ A2 ) @ ( groups2433450451889696826d_enat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5291_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > extended_enat,G: real > extended_enat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_le72135733267957522d_enat @ ( groups2800946370649118462d_enat @ F @ A2 ) @ ( groups2800946370649118462d_enat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5292_sum__strict__mono,axiom,
    ! [A2: set_nat,F: nat > extended_enat,G: nat > extended_enat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_le72135733267957522d_enat @ ( groups7108830773950497114d_enat @ F @ A2 ) @ ( groups7108830773950497114d_enat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5293_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > extended_enat,G: int > extended_enat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_le72135733267957522d_enat @ ( groups4225252721152677374d_enat @ F @ A2 ) @ ( groups4225252721152677374d_enat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5294_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5295_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1935376822645274424al_nat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5296_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5693394587270226106ex_nat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5297_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups4541462559716669496nt_nat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5298_sum_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ X @ A2 )
         => ( ( groups2027974829824023292at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
            = ( groups2027974829824023292at_nat @ G @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ X @ A2 )
         => ( ( groups2027974829824023292at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5299_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1932886352136224148al_int @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5300_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups3539618377306564664at_int @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5301_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5690904116761175830ex_int @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5302_sum_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ X @ A2 )
         => ( ( groups2025484359314973016at_int @ G @ ( insert_Extended_enat @ X @ A2 ) )
            = ( groups2025484359314973016at_int @ G @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ X @ A2 )
         => ( ( groups2025484359314973016at_int @ G @ ( insert_Extended_enat @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups2025484359314973016at_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5303_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( groups8097168146408367636l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5304_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5808333547571424918x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5305_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = zero_zero_nat ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups1935376822645274424al_nat @ G @ S2 )
                        = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5306_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > nat,H2: complex > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
               => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = zero_zero_nat ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups1935376822645274424al_nat @ G @ S2 )
                        = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5307_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > nat,H2: int > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
               => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = zero_zero_nat ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups1935376822645274424al_nat @ G @ S2 )
                        = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5308_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_Extended_enat,S2: set_real,I: extended_enat > real,J: real > extended_enat,T3: set_Extended_enat,G: real > nat,H2: extended_enat > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite4001608067531595151d_enat @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
               => ( member_Extended_enat @ ( J @ A4 ) @ ( minus_925952699566721837d_enat @ T3 @ T5 ) ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: extended_enat] :
                    ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = zero_zero_nat ) )
                 => ( ! [B4: extended_enat] :
                        ( ( member_Extended_enat @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups1935376822645274424al_nat @ G @ S2 )
                        = ( groups2027974829824023292at_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5309_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_real,S2: set_complex,I: real > complex,J: complex > real,T3: set_real,G: complex > nat,H2: real > nat] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = zero_zero_nat ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups5693394587270226106ex_nat @ G @ S2 )
                        = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5310_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_complex,S2: set_complex,I: complex > complex,J: complex > complex,T3: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = zero_zero_nat ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups5693394587270226106ex_nat @ G @ S2 )
                        = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5311_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,G: complex > nat,H2: int > nat] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = zero_zero_nat ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups5693394587270226106ex_nat @ G @ S2 )
                        = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5312_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_Extended_enat,S2: set_complex,I: extended_enat > complex,J: complex > extended_enat,T3: set_Extended_enat,G: complex > nat,H2: extended_enat > nat] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite4001608067531595151d_enat @ T5 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_Extended_enat @ ( J @ A4 ) @ ( minus_925952699566721837d_enat @ T3 @ T5 ) ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: extended_enat] :
                    ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = zero_zero_nat ) )
                 => ( ! [B4: extended_enat] :
                        ( ( member_Extended_enat @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups5693394587270226106ex_nat @ G @ S2 )
                        = ( groups2027974829824023292at_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5313_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > nat,H2: real > nat] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = zero_zero_nat ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups4541462559716669496nt_nat @ G @ S2 )
                        = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5314_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,G: int > nat,H2: complex > nat] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = zero_zero_nat ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups4541462559716669496nt_nat @ G @ S2 )
                        = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5315_round__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim8280529875227126926d_real @ Y ) ) ) ).

% round_mono
thf(fact_5316_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > extended_enat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
       => ( ( ( groups2800946370649118462d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5317_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > extended_enat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
       => ( ( ( groups7108830773950497114d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5318_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > extended_enat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
       => ( ( ( groups1752964319039525884d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5319_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > extended_enat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
       => ( ( ( groups4225252721152677374d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5320_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: extended_enat > extended_enat,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I4: extended_enat] :
            ( ( member_Extended_enat @ I4 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
       => ( ( ( groups2433450451889696826d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_Extended_enat @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5321_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5322_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5323_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5324_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: extended_enat > real,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I4: extended_enat] :
            ( ( member_Extended_enat @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_Extended_enat @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5325_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5326_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > extended_enat,B: extended_enat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
       => ( ( ( groups2800946370649118462d_enat @ F @ S )
            = B )
         => ( ( member_real @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5327_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > extended_enat,B: extended_enat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
       => ( ( ( groups7108830773950497114d_enat @ F @ S )
            = B )
         => ( ( member_nat @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5328_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > extended_enat,B: extended_enat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
       => ( ( ( groups1752964319039525884d_enat @ F @ S )
            = B )
         => ( ( member_complex @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5329_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > extended_enat,B: extended_enat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
       => ( ( ( groups4225252721152677374d_enat @ F @ S )
            = B )
         => ( ( member_int @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5330_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: extended_enat > extended_enat,B: extended_enat,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I4: extended_enat] :
            ( ( member_Extended_enat @ I4 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
       => ( ( ( groups2433450451889696826d_enat @ F @ S )
            = B )
         => ( ( member_Extended_enat @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5331_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B: real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5332_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5333_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5334_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: extended_enat > real,B: real,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I4: extended_enat] :
            ( ( member_Extended_enat @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ S )
            = B )
         => ( ( member_Extended_enat @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5335_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > nat,B: nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = B )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5336_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5337_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5338_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5339_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2027974829824023292at_nat @ G
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X4: extended_enat] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5340_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5341_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5342_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5343_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X4: extended_enat] :
                  ( ( G @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups4148127829035722712t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5344_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1932886352136224148al_int @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = zero_zero_int ) ) ) )
        = ( groups1932886352136224148al_int @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5345_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = zero_zero_int ) ) ) )
        = ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5346_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > extended_enat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups2800946370649118462d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5347_sum__pos2,axiom,
    ! [I6: set_nat,I: nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I4: nat] :
                ( ( member_nat @ I4 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups7108830773950497114d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5348_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I4: complex] :
                ( ( member_complex @ I4 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups1752964319039525884d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5349_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: int > extended_enat] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I4: int] :
                ( ( member_int @ I4 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups4225252721152677374d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5350_sum__pos2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( member_Extended_enat @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups2433450451889696826d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5351_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5352_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I4: complex] :
                ( ( member_complex @ I4 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5353_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I4: int] :
                ( ( member_int @ I4 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5354_sum__pos2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( member_Extended_enat @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5355_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > nat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5356_sum__pos,axiom,
    ! [I6: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5357_sum__pos,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups2027974829824023292at_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5358_sum__pos,axiom,
    ! [I6: set_real,F: real > nat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5359_sum__pos,axiom,
    ! [I6: set_int,F: int > nat] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5360_sum__pos,axiom,
    ! [I6: set_complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
         => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups1752964319039525884d_enat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5361_sum__pos,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I6 )
             => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
         => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups2433450451889696826d_enat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5362_sum__pos,axiom,
    ! [I6: set_real,F: real > extended_enat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
         => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups2800946370649118462d_enat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5363_sum__pos,axiom,
    ! [I6: set_nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I6 )
             => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
         => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups7108830773950497114d_enat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5364_sum__pos,axiom,
    ! [I6: set_int,F: int > extended_enat] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I4 ) ) )
         => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups4225252721152677374d_enat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5365_sum__pos,axiom,
    ! [I6: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5366_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ T3 )
              = ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5367_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
              = ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5368_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2027974829824023292at_nat @ G @ T3 )
              = ( groups2027974829824023292at_nat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5369_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T3 )
              = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5370_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T3 )
              = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5371_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4148127829035722712t_real @ G @ T3 )
              = ( groups4148127829035722712t_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5372_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1932886352136224148al_int @ G @ T3 )
              = ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5373_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5690904116761175830ex_int @ G @ T3 )
              = ( groups5690904116761175830ex_int @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5374_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > int,H2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2025484359314973016at_int @ G @ T3 )
              = ( groups2025484359314973016at_int @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5375_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ T3 )
              = ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5376_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ S2 )
              = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5377_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5693394587270226106ex_nat @ G @ S2 )
              = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5378_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H2: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2027974829824023292at_nat @ G @ S2 )
              = ( groups2027974829824023292at_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5379_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S2 )
              = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5380_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S2 )
              = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5381_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H2: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4148127829035722712t_real @ G @ S2 )
              = ( groups4148127829035722712t_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5382_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_int ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1932886352136224148al_int @ G @ S2 )
              = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5383_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_int ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5690904116761175830ex_int @ G @ S2 )
              = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5384_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H2: extended_enat > int,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_int ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2025484359314973016at_int @ G @ S2 )
              = ( groups2025484359314973016at_int @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5385_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ S2 )
              = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5386_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T3 )
            = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5387_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups2027974829824023292at_nat @ G @ T3 )
            = ( groups2027974829824023292at_nat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5388_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T3 )
            = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5389_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups4148127829035722712t_real @ G @ T3 )
            = ( groups4148127829035722712t_real @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5390_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T3 )
            = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5391_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups2025484359314973016at_int @ G @ T3 )
            = ( groups2025484359314973016at_int @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5392_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups6818542070133387226omplex @ G @ T3 )
            = ( groups6818542070133387226omplex @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5393_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_z5237406670263579293d_enat ) )
         => ( ( groups1752964319039525884d_enat @ G @ T3 )
            = ( groups1752964319039525884d_enat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5394_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_z5237406670263579293d_enat ) )
         => ( ( groups2433450451889696826d_enat @ G @ T3 )
            = ( groups2433450451889696826d_enat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5395_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ T3 )
            = ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5396_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S2 )
            = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5397_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups2027974829824023292at_nat @ G @ S2 )
            = ( groups2027974829824023292at_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5398_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S2 )
            = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5399_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups4148127829035722712t_real @ G @ S2 )
            = ( groups4148127829035722712t_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5400_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S2 )
            = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5401_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups2025484359314973016at_int @ G @ S2 )
            = ( groups2025484359314973016at_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5402_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups6818542070133387226omplex @ G @ S2 )
            = ( groups6818542070133387226omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5403_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_z5237406670263579293d_enat ) )
         => ( ( groups1752964319039525884d_enat @ G @ S2 )
            = ( groups1752964319039525884d_enat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5404_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_z5237406670263579293d_enat ) )
         => ( ( groups2433450451889696826d_enat @ G @ S2 )
            = ( groups2433450451889696826d_enat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5405_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ S2 )
            = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5406_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
               => ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5407_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
               => ( ( groups5693394587270226106ex_nat @ G @ A2 )
                  = ( groups5693394587270226106ex_nat @ H2 @ B ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5408_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups2027974829824023292at_nat @ G @ C4 )
                  = ( groups2027974829824023292at_nat @ H2 @ C4 ) )
               => ( ( groups2027974829824023292at_nat @ G @ A2 )
                  = ( groups2027974829824023292at_nat @ H2 @ B ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5409_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5410_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5411_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups4148127829035722712t_real @ G @ C4 )
                  = ( groups4148127829035722712t_real @ H2 @ C4 ) )
               => ( ( groups4148127829035722712t_real @ G @ A2 )
                  = ( groups4148127829035722712t_real @ H2 @ B ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5412_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups1932886352136224148al_int @ G @ C4 )
                  = ( groups1932886352136224148al_int @ H2 @ C4 ) )
               => ( ( groups1932886352136224148al_int @ G @ A2 )
                  = ( groups1932886352136224148al_int @ H2 @ B ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5413_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups5690904116761175830ex_int @ G @ C4 )
                  = ( groups5690904116761175830ex_int @ H2 @ C4 ) )
               => ( ( groups5690904116761175830ex_int @ G @ A2 )
                  = ( groups5690904116761175830ex_int @ H2 @ B ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5414_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > int,H2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups2025484359314973016at_int @ G @ C4 )
                  = ( groups2025484359314973016at_int @ H2 @ C4 ) )
               => ( ( groups2025484359314973016at_int @ G @ A2 )
                  = ( groups2025484359314973016at_int @ H2 @ B ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5415_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_complex ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) )
               => ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5416_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B ) )
                = ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5417_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
                  = ( groups5693394587270226106ex_nat @ H2 @ B ) )
                = ( ( groups5693394587270226106ex_nat @ G @ C4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5418_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups2027974829824023292at_nat @ G @ A2 )
                  = ( groups2027974829824023292at_nat @ H2 @ B ) )
                = ( ( groups2027974829824023292at_nat @ G @ C4 )
                  = ( groups2027974829824023292at_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5419_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5420_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5421_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups4148127829035722712t_real @ G @ A2 )
                  = ( groups4148127829035722712t_real @ H2 @ B ) )
                = ( ( groups4148127829035722712t_real @ G @ C4 )
                  = ( groups4148127829035722712t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5422_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups1932886352136224148al_int @ G @ A2 )
                  = ( groups1932886352136224148al_int @ H2 @ B ) )
                = ( ( groups1932886352136224148al_int @ G @ C4 )
                  = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5423_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
                  = ( groups5690904116761175830ex_int @ H2 @ B ) )
                = ( ( groups5690904116761175830ex_int @ G @ C4 )
                  = ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5424_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > int,H2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups2025484359314973016at_int @ G @ A2 )
                  = ( groups2025484359314973016at_int @ H2 @ B ) )
                = ( ( groups2025484359314973016at_int @ G @ C4 )
                  = ( groups2025484359314973016at_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5425_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_complex ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B ) )
                = ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5426_sum_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B ) ) @ ( groups5693394587270226106ex_nat @ G @ B ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5427_sum_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > nat] :
      ( ( ord_le7203529160286727270d_enat @ B @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2027974829824023292at_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ B ) ) @ ( groups2027974829824023292at_nat @ G @ B ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5428_sum_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B ) ) @ ( groups5690904116761175830ex_int @ G @ B ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5429_sum_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > int] :
      ( ( ord_le7203529160286727270d_enat @ B @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2025484359314973016at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups2025484359314973016at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ B ) ) @ ( groups2025484359314973016at_int @ G @ B ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5430_sum_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B ) ) @ ( groups5808333547571424918x_real @ G @ B ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5431_sum_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > real] :
      ( ( ord_le7203529160286727270d_enat @ B @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups4148127829035722712t_real @ G @ A2 )
          = ( plus_plus_real @ ( groups4148127829035722712t_real @ G @ ( minus_925952699566721837d_enat @ A2 @ B ) ) @ ( groups4148127829035722712t_real @ G @ B ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5432_sum_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: complex > extended_enat] :
      ( ( ord_le211207098394363844omplex @ B @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups1752964319039525884d_enat @ G @ A2 )
          = ( plus_p3455044024723400733d_enat @ ( groups1752964319039525884d_enat @ G @ ( minus_811609699411566653omplex @ A2 @ B ) ) @ ( groups1752964319039525884d_enat @ G @ B ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5433_sum_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ B @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2433450451889696826d_enat @ G @ A2 )
          = ( plus_p3455044024723400733d_enat @ ( groups2433450451889696826d_enat @ G @ ( minus_925952699566721837d_enat @ A2 @ B ) ) @ ( groups2433450451889696826d_enat @ G @ B ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5434_sum_Osubset__diff,axiom,
    ! [B: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B ) ) @ ( groups3539618377306564664at_int @ G @ B ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5435_sum_Osubset__diff,axiom,
    ! [B: set_nat,A2: set_nat,G: nat > extended_enat] :
      ( ( ord_less_eq_set_nat @ B @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups7108830773950497114d_enat @ G @ A2 )
          = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( minus_minus_set_nat @ A2 @ B ) ) @ ( groups7108830773950497114d_enat @ G @ B ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5436_sum__diff,axiom,
    ! [A2: set_complex,B: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B @ A2 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B ) ) ) ) ) ).

% sum_diff
thf(fact_5437_sum__diff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ord_le7203529160286727270d_enat @ B @ A2 )
       => ( ( groups2025484359314973016at_int @ F @ ( minus_925952699566721837d_enat @ A2 @ B ) )
          = ( minus_minus_int @ ( groups2025484359314973016at_int @ F @ A2 ) @ ( groups2025484359314973016at_int @ F @ B ) ) ) ) ) ).

% sum_diff
thf(fact_5438_sum__diff,axiom,
    ! [A2: set_complex,B: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B @ A2 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B ) ) ) ) ) ).

% sum_diff
thf(fact_5439_sum__diff,axiom,
    ! [A2: set_Extended_enat,B: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ord_le7203529160286727270d_enat @ B @ A2 )
       => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A2 @ B ) )
          = ( minus_minus_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( groups4148127829035722712t_real @ F @ B ) ) ) ) ) ).

% sum_diff
thf(fact_5440_sum__diff,axiom,
    ! [A2: set_nat,B: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B @ A2 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B ) ) ) ) ) ).

% sum_diff
thf(fact_5441_sum__diff,axiom,
    ! [A2: set_int,B: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B @ A2 )
       => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B ) )
          = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B ) ) ) ) ) ).

% sum_diff
thf(fact_5442_sum__diff,axiom,
    ! [A2: set_int,B: set_int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B @ A2 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B ) ) ) ) ) ).

% sum_diff
thf(fact_5443_sum__diff,axiom,
    ! [A2: set_complex,B: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B @ A2 )
       => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B ) )
          = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B ) ) ) ) ) ).

% sum_diff
thf(fact_5444_sum__diff,axiom,
    ! [A2: set_nat,B: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B @ A2 )
       => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B ) )
          = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B ) ) ) ) ) ).

% sum_diff
thf(fact_5445_sum__mono2,axiom,
    ! [B: set_real,A2: set_real,F: real > extended_enat] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B @ A2 ) )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ B4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( groups2800946370649118462d_enat @ F @ A2 ) @ ( groups2800946370649118462d_enat @ F @ B ) ) ) ) ) ).

% sum_mono2
thf(fact_5446_sum__mono2,axiom,
    ! [B: set_complex,A2: set_complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B @ A2 ) )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ B4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( groups1752964319039525884d_enat @ F @ A2 ) @ ( groups1752964319039525884d_enat @ F @ B ) ) ) ) ) ).

% sum_mono2
thf(fact_5447_sum__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B @ A2 ) )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ B4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( groups2433450451889696826d_enat @ F @ A2 ) @ ( groups2433450451889696826d_enat @ F @ B ) ) ) ) ) ).

% sum_mono2
thf(fact_5448_sum__mono2,axiom,
    ! [B: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B ) ) ) ) ) ).

% sum_mono2
thf(fact_5449_sum__mono2,axiom,
    ! [B: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B ) ) ) ) ) ).

% sum_mono2
thf(fact_5450_sum__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( groups4148127829035722712t_real @ F @ B ) ) ) ) ) ).

% sum_mono2
thf(fact_5451_sum__mono2,axiom,
    ! [B: set_real,A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B ) ) ) ) ) ).

% sum_mono2
thf(fact_5452_sum__mono2,axiom,
    ! [B: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
         => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B ) ) ) ) ) ).

% sum_mono2
thf(fact_5453_sum__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
         => ( ord_less_eq_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ F @ B ) ) ) ) ) ).

% sum_mono2
thf(fact_5454_sum__mono2,axiom,
    ! [B: set_real,A2: set_real,F: real > int] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B @ A2 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
         => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B ) ) ) ) ) ).

% sum_mono2
thf(fact_5455_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5456_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5457_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5458_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups1752964319039525884d_enat @ G @ A2 )
          = ( plus_p3455044024723400733d_enat @ ( G @ X ) @ ( groups1752964319039525884d_enat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5459_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2027974829824023292at_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups2027974829824023292at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5460_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2025484359314973016at_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups2025484359314973016at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5461_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups4148127829035722712t_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups4148127829035722712t_real @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5462_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2433450451889696826d_enat @ G @ A2 )
          = ( plus_p3455044024723400733d_enat @ ( G @ X ) @ ( groups2433450451889696826d_enat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5463_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups1935376822645274424al_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5464_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups1932886352136224148al_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5465_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5466_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > int,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5467_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5468_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > extended_enat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups1752964319039525884d_enat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ X ) @ ( groups1752964319039525884d_enat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5469_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2027974829824023292at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups2027974829824023292at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5470_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > int,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2025484359314973016at_int @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups2025484359314973016at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5471_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > real,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups4148127829035722712t_real @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5472_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > extended_enat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2433450451889696826d_enat @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ X ) @ ( groups2433450451889696826d_enat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5473_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > nat,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5474_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > int,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5475_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5476_sum__diff1,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ A @ A2 )
         => ( ( groups2025484359314973016at_int @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
            = ( minus_minus_int @ ( groups2025484359314973016at_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ A2 )
         => ( ( groups2025484359314973016at_int @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
            = ( groups2025484359314973016at_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5477_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1932886352136224148al_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5478_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5479_sum__diff1,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ A @ A2 )
         => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
            = ( minus_minus_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ A2 )
         => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
            = ( groups4148127829035722712t_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5480_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5481_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5482_sum__diff1,axiom,
    ! [A2: set_nat,A: nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ A @ A2 )
         => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
            = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_nat @ A @ A2 )
         => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
            = ( groups3539618377306564664at_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5483_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5484_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups7754918857620584856omplex @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5485_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B2 @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5486_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_int @ ( B2 @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5487_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_real @ ( B2 @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5488_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: complex > extended_enat,C: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups1752964319039525884d_enat
              @ ^ [K2: complex] : ( if_Extended_enat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_p3455044024723400733d_enat @ ( B2 @ A ) @ ( groups1752964319039525884d_enat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups1752964319039525884d_enat
              @ ^ [K2: complex] : ( if_Extended_enat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups1752964319039525884d_enat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5489_sum_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > nat,C: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B2 @ A ) @ ( groups2027974829824023292at_nat @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups2027974829824023292at_nat @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5490_sum_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > int,C: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2025484359314973016at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_int @ ( B2 @ A ) @ ( groups2025484359314973016at_int @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2025484359314973016at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups2025484359314973016at_int @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5491_sum_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > real,C: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_real @ ( B2 @ A ) @ ( groups4148127829035722712t_real @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups4148127829035722712t_real @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5492_sum_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > extended_enat,C: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2433450451889696826d_enat
              @ ^ [K2: extended_enat] : ( if_Extended_enat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_p3455044024723400733d_enat @ ( B2 @ A ) @ ( groups2433450451889696826d_enat @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2433450451889696826d_enat
              @ ^ [K2: extended_enat] : ( if_Extended_enat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups2433450451889696826d_enat @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5493_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B2: real > nat,C: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B2 @ A ) @ ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5494_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B2: real > int,C: real > int] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_int @ ( B2 @ A ) @ ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5495_sum__strict__mono2,axiom,
    ! [B: set_real,A2: set_real,B2: real,F: real > real] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ( member_real @ B2 @ ( minus_minus_set_real @ B @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B2 ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5496_sum__strict__mono2,axiom,
    ! [B: set_complex,A2: set_complex,B2: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B2 ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5497_sum__strict__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,B2: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( ( member_Extended_enat @ B2 @ ( minus_925952699566721837d_enat @ B @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B2 ) )
           => ( ! [X5: extended_enat] :
                  ( ( member_Extended_enat @ X5 @ B )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( groups4148127829035722712t_real @ F @ B ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5498_sum__strict__mono2,axiom,
    ! [B: set_real,A2: set_real,B2: real,F: real > nat] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ( member_real @ B2 @ ( minus_minus_set_real @ B @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B2 ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5499_sum__strict__mono2,axiom,
    ! [B: set_complex,A2: set_complex,B2: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B2 ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5500_sum__strict__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,B2: extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( ( member_Extended_enat @ B2 @ ( minus_925952699566721837d_enat @ B @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B2 ) )
           => ( ! [X5: extended_enat] :
                  ( ( member_Extended_enat @ X5 @ B )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ F @ B ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5501_sum__strict__mono2,axiom,
    ! [B: set_real,A2: set_real,B2: real,F: real > int] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ( member_real @ B2 @ ( minus_minus_set_real @ B @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
             => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5502_sum__strict__mono2,axiom,
    ! [B: set_complex,A2: set_complex,B2: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
             => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5503_sum__strict__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,B2: extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( ( member_Extended_enat @ B2 @ ( minus_925952699566721837d_enat @ B @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
           => ( ! [X5: extended_enat] :
                  ( ( member_Extended_enat @ X5 @ B )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
             => ( ord_less_int @ ( groups2025484359314973016at_int @ F @ A2 ) @ ( groups2025484359314973016at_int @ F @ B ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5504_sum__strict__mono2,axiom,
    ! [B: set_nat,A2: set_nat,B2: nat,F: nat > int] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
           => ( ! [X5: nat] :
                  ( ( member_nat @ X5 @ B )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
             => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5505_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > extended_enat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups1752964319039525884d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5506_member__le__sum,axiom,
    ! [I: extended_enat,A2: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( member_Extended_enat @ I @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite4001608067531595151d_enat @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups2433450451889696826d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5507_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > extended_enat] :
      ( ( member_real @ I @ A2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups2800946370649118462d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5508_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > extended_enat] :
      ( ( member_int @ I @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups4225252721152677374d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5509_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: nat > extended_enat] :
      ( ( member_nat @ I @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite_finite_nat @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups7108830773950497114d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5510_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5511_member__le__sum,axiom,
    ! [I: extended_enat,A2: set_Extended_enat,F: extended_enat > real] :
      ( ( member_Extended_enat @ I @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite4001608067531595151d_enat @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups4148127829035722712t_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5512_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > real] :
      ( ( member_real @ I @ A2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5513_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > real] :
      ( ( member_int @ I @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5514_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > nat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_nat @ ( F @ I ) @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5515_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_5516_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_5517_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_5518_Sum__Icc__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_5519_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_5520_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_5521_arith__series__nat,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_5522_bit__0__eq,axiom,
    ( ( bit_se1146084159140164899it_int @ zero_zero_int )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_5523_bit__0__eq,axiom,
    ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_5524_ceiling__zero,axiom,
    ( ( archim7802044766580827645g_real @ zero_zero_real )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_5525_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_5526_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_5527_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M2: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M2 ) @ N2 ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_5528_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M2: num,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( suc @ N2 ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M2 ) @ N2 ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_5529_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M2: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M2 ) @ N2 ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_5530_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M2: num,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( suc @ N2 ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M2 ) @ N2 ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_5531_ceiling__add__of__int,axiom,
    ! [X: real,Z3: int] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z3 ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z3 ) ) ).

% ceiling_add_of_int
thf(fact_5532_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_5533_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_5534_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_5535_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_5536_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_5537_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_5538_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_5539_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_5540_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_5541_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_5542_ceiling__add__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_5543_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_5544_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N2 ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_5545_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N2 ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_5546_bit__0,axiom,
    ! [A: int] :
      ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_5547_bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_5548_and__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se727722235901077358nd_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_5549_Suc__0__and__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_5550_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5551_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5552_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > extended_enat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_z5237406670263579293d_enat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5553_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5554_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5555_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_5556_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_5557_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_5558_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_5559_bit__mod__2__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N2 )
      = ( ( N2 = zero_zero_nat )
        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_5560_bit__mod__2__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
      = ( ( N2 = zero_zero_nat )
        & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_5561_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_5562_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_5563_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_5564_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_5565_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
            @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_5566_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
            @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_5567_bit__disjunctive__add__iff,axiom,
    ! [A: int,B2: int,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N3 )
          | ~ ( bit_se1146084159140164899it_int @ B2 @ N3 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B2 ) @ N2 )
        = ( ( bit_se1146084159140164899it_int @ A @ N2 )
          | ( bit_se1146084159140164899it_int @ B2 @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_5568_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B2: nat,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N3 )
          | ~ ( bit_se1148574629649215175it_nat @ B2 @ N3 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B2 ) @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ( bit_se1148574629649215175it_nat @ B2 @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_5569_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_5570_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_5571_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1146084159140164899it_int @ one_one_int @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_5572_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_5573_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ ( suc @ X5 ) @ A2 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G @ ( suc @ X5 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_5574_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ ( suc @ X5 ) @ A2 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G @ ( suc @ X5 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A2 )
          = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_5575_bit__take__bit__iff,axiom,
    ! [M2: nat,A: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M2 @ A ) @ N2 )
      = ( ( ord_less_nat @ N2 @ M2 )
        & ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).

% bit_take_bit_iff
thf(fact_5576_bit__take__bit__iff,axiom,
    ! [M2: nat,A: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M2 @ A ) @ N2 )
      = ( ( ord_less_nat @ N2 @ M2 )
        & ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).

% bit_take_bit_iff
thf(fact_5577_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_5578_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_5579_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_5580_bit__of__bool__iff,axiom,
    ! [B2: $o,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ N2 )
      = ( B2
        & ( N2 = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_5581_bit__of__bool__iff,axiom,
    ! [B2: $o,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ N2 )
      = ( B2
        & ( N2 = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_5582_sum__subtractf__nat,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat,F: extended_enat > nat] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups2027974829824023292at_nat
          @ ^ [X4: extended_enat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5583_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: real > nat,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X4: real] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5584_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X4: set_nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5585_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: int > nat,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5586_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5587_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_5588_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M2: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_5589_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M2: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_5590_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M2: nat,K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_5591_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y5: complex] :
                  ( ( member_complex @ Y5 @ A2 )
                 => ( ( X4 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_5592_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y5: int] :
                  ( ( member_int @ Y5 @ A2 )
                 => ( ( X4 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_5593_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2027974829824023292at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y5: extended_enat] :
                  ( ( member_Extended_enat @ Y5 @ A2 )
                 => ( ( X4 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_5594_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y5: nat] :
                  ( ( member_nat @ Y5 @ A2 )
                 => ( ( X4 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_5595_sum__SucD,axiom,
    ! [F: nat > nat,A2: set_nat,N2: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A2 )
        = ( suc @ N2 ) )
     => ? [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).

% sum_SucD
thf(fact_5596_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ( F @ X4 )
                = one_one_nat )
              & ! [Y5: complex] :
                  ( ( member_complex @ Y5 @ A2 )
                 => ( ( X4 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5597_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ( F @ X4 )
                = one_one_nat )
              & ! [Y5: int] :
                  ( ( member_int @ Y5 @ A2 )
                 => ( ( X4 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5598_sum__eq__1__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2027974829824023292at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
              & ( ( F @ X4 )
                = one_one_nat )
              & ! [Y5: extended_enat] :
                  ( ( member_Extended_enat @ Y5 @ A2 )
                 => ( ( X4 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5599_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( ( F @ X4 )
                = one_one_nat )
              & ! [Y5: nat] :
                  ( ( member_nat @ Y5 @ A2 )
                 => ( ( X4 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5600_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_5601_ceiling__le__iff,axiom,
    ! [X: real,Z3: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z3 )
      = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ) ).

% ceiling_le_iff
thf(fact_5602_less__ceiling__iff,axiom,
    ! [Z3: int,X: real] :
      ( ( ord_less_int @ Z3 @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ) ).

% less_ceiling_iff
thf(fact_5603_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_5604_sum__power__add,axiom,
    ! [X: int,M2: nat,I6: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M2 @ I3 ) )
        @ I6 )
      = ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_5605_sum__power__add,axiom,
    ! [X: complex,M2: nat,I6: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M2 @ I3 ) )
        @ I6 )
      = ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_5606_sum__power__add,axiom,
    ! [X: real,M2: nat,I6: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M2 @ I3 ) )
        @ I6 )
      = ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_5607_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_5608_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N2: nat,M2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_5609_sum__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X4: complex] : X4
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_5610_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X4: complex] : X4
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_5611_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_5612_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_5613_sum__diff__nat,axiom,
    ! [B: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ B @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B ) ) ) ) ) ).

% sum_diff_nat
thf(fact_5614_sum__diff__nat,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ B @ A2 )
       => ( ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ A2 @ B ) )
          = ( minus_minus_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ F @ B ) ) ) ) ) ).

% sum_diff_nat
thf(fact_5615_sum__diff__nat,axiom,
    ! [B: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B )
     => ( ( ord_less_eq_set_int @ B @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B ) ) ) ) ) ).

% sum_diff_nat
thf(fact_5616_sum__diff__nat,axiom,
    ! [B: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ B @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B ) ) ) ) ) ).

% sum_diff_nat
thf(fact_5617_sum__diff1__nat,axiom,
    ! [A: set_nat,A2: set_set_nat,F: set_nat > nat] :
      ( ( ( member_set_nat @ A @ A2 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_set_nat @ A @ A2 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
          = ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_5618_sum__diff1__nat,axiom,
    ! [A: extended_enat,A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( ( member_Extended_enat @ A @ A2 )
       => ( ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
          = ( minus_minus_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_Extended_enat @ A @ A2 )
       => ( ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
          = ( groups2027974829824023292at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_5619_sum__diff1__nat,axiom,
    ! [A: real,A2: set_real,F: real > nat] :
      ( ( ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_5620_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: int > nat] :
      ( ( ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_5621_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: nat > nat] :
      ( ( ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_5622_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5623_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5624_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > extended_enat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_z5237406670263579293d_enat )
     => ( ( groups7108830773950497114d_enat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups7108830773950497114d_enat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5625_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5626_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > real,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5627_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5628_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5629_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5630_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5631_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5632_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ ( suc @ N2 ) ) @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5633_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5634_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5635_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_plus_int @ ( G @ M2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5636_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ M2 ) @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5637_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_plus_nat @ ( G @ M2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5638_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_plus_real @ ( G @ M2 ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5639_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M2: nat,K: int] :
      ( ( ord_less_nat @ N2 @ M2 )
     => ( ( bit_se1146084159140164899it_int @ K @ N2 )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M2 @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_5640_bit__concat__bit__iff,axiom,
    ! [M2: nat,K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M2 @ K @ L ) @ N2 )
      = ( ( ( ord_less_nat @ N2 @ M2 )
          & ( bit_se1146084159140164899it_int @ K @ N2 ) )
        | ( ( ord_less_eq_nat @ M2 @ N2 )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_5641_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ M2 )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5642_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ M2 )
          @ ( groups7108830773950497114d_enat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5643_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ M2 )
          @ ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5644_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ M2 )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5645_sum__Suc__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_5646_sum__Suc__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_5647_exp__eq__0__imp__not__bit,axiom,
    ! [N2: nat,A: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
        = zero_zero_int )
     => ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_5648_exp__eq__0__imp__not__bit,axiom,
    ! [N2: nat,A: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        = zero_zero_nat )
     => ~ ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_5649_bit__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N2 ) ) ).

% bit_Suc
thf(fact_5650_bit__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N2 ) )
      = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 ) ) ).

% bit_Suc
thf(fact_5651_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_5652_ceiling__unique,axiom,
    ! [Z3: int,X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z3 ) @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) )
       => ( ( archim7802044766580827645g_real @ X )
          = Z3 ) ) ) ).

% ceiling_unique
thf(fact_5653_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_5654_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I3 ) ) )
           => ( P @ I3 ) ) ) ) ).

% ceiling_split
thf(fact_5655_mult__ceiling__le,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B2 ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_5656_ceiling__less__iff,axiom,
    ! [X: real,Z3: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z3 )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z3 ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_5657_le__ceiling__iff,axiom,
    ! [Z3: int,X: real] :
      ( ( ord_less_eq_int @ Z3 @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z3 ) @ one_one_real ) @ X ) ) ).

% le_ceiling_iff
thf(fact_5658_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_eq_nat @ N3 @ M5 )
             => ( ( bit_se1146084159140164899it_int @ K @ M5 )
                = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).

% int_bit_bound
thf(fact_5659_sum_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > int,P5: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P5 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P5 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5660_sum_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > extended_enat,P5: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P5 ) ) )
        = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P5 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5661_sum_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > nat,P5: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P5 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P5 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5662_sum_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > real,P5: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P5 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P5 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5663_sum__count__set,axiom,
    ! [Xs: list_complex,X8: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X8 )
     => ( ( finite3207457112153483333omplex @ X8 )
       => ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X8 )
          = ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_5664_sum__count__set,axiom,
    ! [Xs: list_Extended_enat,X8: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs ) @ X8 )
     => ( ( finite4001608067531595151d_enat @ X8 )
       => ( ( groups2027974829824023292at_nat @ ( count_101369445342291426d_enat @ Xs ) @ X8 )
          = ( size_s3941691890525107288d_enat @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_5665_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X8: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X8 )
     => ( ( finite5795047828879050333T_VEBT @ X8 )
       => ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X8 )
          = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_5666_sum__count__set,axiom,
    ! [Xs: list_int,X8: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X8 )
     => ( ( finite_finite_int @ X8 )
       => ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X8 )
          = ( size_size_list_int @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_5667_sum__count__set,axiom,
    ! [Xs: list_nat,X8: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X8 )
     => ( ( finite_finite_nat @ X8 )
       => ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X8 )
          = ( size_size_list_nat @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_5668_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,N2: nat] :
      ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
        = zero_zero_int )
      = ( ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_5669_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
        = zero_zero_nat )
      = ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_5670_ceiling__divide__upper,axiom,
    ! [Q3: real,P5: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ P5 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_5671_sum__natinterval__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > complex] :
      ( ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( minus_minus_complex @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_5672_sum__natinterval__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( minus_minus_int @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_5673_sum__natinterval__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( minus_minus_real @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_5674_sum__telescope_H_H,axiom,
    ! [M2: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_5675_sum__telescope_H_H,axiom,
    ! [M2: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) )
        = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_5676_even__bit__succ__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N2 )
        = ( ( bit_se1146084159140164899it_int @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_5677_even__bit__succ__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_5678_odd__bit__iff__bit__pred,axiom,
    ! [A: int,N2: nat] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ A @ N2 )
        = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_5679_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ A @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_5680_ceiling__divide__lower,axiom,
    ! [Q3: real,P5: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P5 ) ) ).

% ceiling_divide_lower
thf(fact_5681_ceiling__eq,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X )
     => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X )
          = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_5682_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_5683_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_5684_sum__gp__multiplied,axiom,
    ! [M2: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5685_sum__gp__multiplied,axiom,
    ! [M2: nat,N2: nat,X: complex] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X @ M2 ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5686_sum__gp__multiplied,axiom,
    ! [M2: nat,N2: nat,X: real] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) )
        = ( minus_minus_real @ ( power_power_real @ X @ M2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5687_sum_Oin__pairs,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_5688_sum_Oin__pairs,axiom,
    ! [G: nat > extended_enat,M2: nat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups7108830773950497114d_enat
        @ ^ [I3: nat] : ( plus_p3455044024723400733d_enat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_5689_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_5690_sum_Oin__pairs,axiom,
    ! [G: nat > real,M2: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_5691_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K3: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
           => ( ( ~ ( ( member_int @ K3 @ ( 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 @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K3 @ L4 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_5692_bit__sum__mult__2__cases,axiom,
    ! [A: int,B2: int,N2: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J3 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_5693_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B2: nat,N2: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J3 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_5694_bit__rec,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A3: int,N: nat] :
          ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_5695_bit__rec,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A3: nat,N: nat] :
          ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_5696_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M: nat,N: nat] :
          ( if_nat
          @ ( ( M = zero_zero_nat )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_5697_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_5698_mask__eq__sum__exp__nat,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_5699_gauss__sum__nat,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_5700_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_5701_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I4: int,J3: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I4 @ J3 ) )
           => ( ( ( ord_less_eq_int @ I4 @ J3 )
               => ( P @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) )
             => ( P @ I4 @ J3 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_5702_sum__gp,axiom,
    ! [N2: nat,M2: nat,X: complex] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M2 ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5703_sum__gp,axiom,
    ! [N2: nat,M2: nat,X: real] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5704_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_5705_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_5706_sum__gp__offset,axiom,
    ! [X: complex,M2: nat,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N2 ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N2 ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_5707_sum__gp__offset,axiom,
    ! [X: real,M2: nat,N2: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N2 ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N2 ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_5708_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_5709_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_5710_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_5711_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_5712_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_5713_arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_5714_arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_5715_of__nat__eq__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M2 )
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( M2 = N2 ) ) ).

% of_nat_eq_iff
thf(fact_5716_of__nat__eq__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M2 = N2 ) ) ).

% of_nat_eq_iff
thf(fact_5717_of__nat__eq__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M2 )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M2 = N2 ) ) ).

% of_nat_eq_iff
thf(fact_5718_numeral__le__real__of__nat__iff,axiom,
    ! [N2: num,M2: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M2 ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_5719_int__eq__iff__numeral,axiom,
    ! [M2: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( numeral_numeral_int @ V ) )
      = ( M2
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_5720_negative__eq__positive,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
        = ( semiri1314217659103216013at_int @ M2 ) )
      = ( ( N2 = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_5721_of__int__of__nat__eq,axiom,
    ! [N2: nat] :
      ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% of_int_of_nat_eq
thf(fact_5722_of__int__of__nat__eq,axiom,
    ! [N2: nat] :
      ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% of_int_of_nat_eq
thf(fact_5723_negative__zle,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).

% negative_zle
thf(fact_5724_int__dvd__int__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( dvd_dvd_nat @ M2 @ N2 ) ) ).

% int_dvd_int_iff
thf(fact_5725_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_5726_of__nat__0,axiom,
    ( ( semiri4216267220026989637d_enat @ zero_zero_nat )
    = zero_z5237406670263579293d_enat ) ).

% of_nat_0
thf(fact_5727_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_5728_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_5729_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_5730_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_5731_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_z5237406670263579293d_enat
        = ( semiri4216267220026989637d_enat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_5732_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_5733_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_5734_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_5735_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri8010041392384452111omplex @ M2 )
        = zero_zero_complex )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_5736_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri4216267220026989637d_enat @ M2 )
        = zero_z5237406670263579293d_enat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_5737_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_5738_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_5739_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_5740_of__nat__less__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M2 ) @ ( semiri4216267220026989637d_enat @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% of_nat_less_iff
thf(fact_5741_of__nat__less__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% of_nat_less_iff
thf(fact_5742_of__nat__less__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% of_nat_less_iff
thf(fact_5743_of__nat__less__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% of_nat_less_iff
thf(fact_5744_of__nat__le__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% of_nat_le_iff
thf(fact_5745_of__nat__le__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% of_nat_le_iff
thf(fact_5746_of__nat__le__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% of_nat_le_iff
thf(fact_5747_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N2: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N2 ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_5748_real__of__nat__less__numeral__iff,axiom,
    ! [N2: nat,W2: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W2 ) )
      = ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W2 ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_5749_of__nat__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri4216267220026989637d_enat @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ M2 ) @ ( semiri4216267220026989637d_enat @ N2 ) ) ) ).

% of_nat_add
thf(fact_5750_of__nat__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_add
thf(fact_5751_of__nat__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_add
thf(fact_5752_of__nat__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_add
thf(fact_5753_of__nat__mult,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M2 @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).

% of_nat_mult
thf(fact_5754_of__nat__mult,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri4216267220026989637d_enat @ ( times_times_nat @ M2 @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ M2 ) @ ( semiri4216267220026989637d_enat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_5755_of__nat__mult,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_mult
thf(fact_5756_of__nat__mult,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mult
thf(fact_5757_of__nat__mult,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_5758_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ N2 )
        = one_one_complex )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5759_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ N2 )
        = one_one_real )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5760_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ N2 )
        = one_one_int )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5761_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N2 )
        = one_one_nat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5762_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5763_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5764_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5765_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5766_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_5767_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_5768_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_5769_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_5770_negative__zless,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).

% negative_zless
thf(fact_5771_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_5772_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_5773_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_5774_of__nat__sum,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_5775_of__nat__sum,axiom,
    ! [F: complex > nat,A2: set_complex] :
      ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X4: complex] : ( semiri8010041392384452111omplex @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_5776_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_5777_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( semiri1316708129612266289at_nat @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_5778_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X4: nat] : ( semiri5074537144036343181t_real @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_5779_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( semiri4216267220026989637d_enat @ M2 ) @ zero_z5237406670263579293d_enat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_5780_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_5781_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_5782_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_5783_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M2 ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_5784_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri4216267220026989637d_enat @ ( suc @ M2 ) )
      = ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( semiri4216267220026989637d_enat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_5785_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M2 ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_5786_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_5787_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_5788_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( semiri4216267220026989637d_enat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_5789_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_5790_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_5791_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_5792_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_5793_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_5794_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_5795_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_5796_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_5797_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_5798_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_5799_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_5800_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_5801_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_5802_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_5803_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_5804_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_5805_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_5806_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_5807_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5808_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5809_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5810_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5811_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5812_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5813_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5814_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5815_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5816_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5817_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5818_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5819_real__arch__simple,axiom,
    ! [X: real] :
    ? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% real_arch_simple
thf(fact_5820_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% reals_Archimedean2
thf(fact_5821_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_5822_mult__of__nat__commute,axiom,
    ! [X: nat,Y: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ X ) @ Y )
      = ( times_7803423173614009249d_enat @ Y @ ( semiri4216267220026989637d_enat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_5823_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_5824_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_5825_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_5826_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat,M: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% nat_less_real_le
thf(fact_5827_int__cases2,axiom,
    ! [Z3: int] :
      ( ! [N3: nat] :
          ( Z3
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( Z3
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% int_cases2
thf(fact_5828_int__diff__cases,axiom,
    ! [Z3: int] :
      ~ ! [M3: nat,N3: nat] :
          ( Z3
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_diff_cases
thf(fact_5829_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5830_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5831_bit__Suc__0__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_5832_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_5833_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( semiri4216267220026989637d_enat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_5834_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_5835_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_5836_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_5837_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M2 ) @ zero_z5237406670263579293d_enat ) ).

% of_nat_less_0_iff
thf(fact_5838_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_5839_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_5840_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_5841_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_5842_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri4216267220026989637d_enat @ ( suc @ N2 ) )
     != zero_z5237406670263579293d_enat ) ).

% of_nat_neq_0
thf(fact_5843_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_5844_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_5845_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_5846_less__imp__of__nat__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M2 ) @ ( semiri4216267220026989637d_enat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_5847_less__imp__of__nat__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_5848_less__imp__of__nat__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_5849_less__imp__of__nat__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_5850_of__nat__less__imp__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M2 ) @ ( semiri4216267220026989637d_enat @ N2 ) )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_5851_of__nat__less__imp__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_5852_of__nat__less__imp__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_5853_of__nat__less__imp__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_5854_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_5855_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_5856_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_5857_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N: nat,M: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_5858_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_5859_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_5860_int__cases,axiom,
    ! [Z3: int] :
      ( ! [N3: nat] :
          ( Z3
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( Z3
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% int_cases
thf(fact_5861_int__of__nat__induct,axiom,
    ! [P: int > $o,Z3: int] :
      ( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
     => ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
       => ( P @ Z3 ) ) ) ).

% int_of_nat_induct
thf(fact_5862_zle__int,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% zle_int
thf(fact_5863_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_5864_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N3: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% nonneg_int_cases
thf(fact_5865_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N3: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_5866_zadd__int__left,axiom,
    ! [M2: nat,N2: nat,Z3: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z3 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) ) @ Z3 ) ) ).

% zadd_int_left
thf(fact_5867_int__plus,axiom,
    ! [N2: nat,M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% int_plus
thf(fact_5868_int__ops_I5_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% int_ops(5)
thf(fact_5869_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z6: int] :
        ? [N: nat] :
          ( Z6
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_5870_not__int__zless__negative,axiom,
    ! [N2: nat,M2: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% not_int_zless_negative
thf(fact_5871_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_5872_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_5873_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_5874_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_less_as_int
thf(fact_5875_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_leq_as_int
thf(fact_5876_not__bit__Suc__0__numeral,axiom,
    ! [N2: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N2 ) ) ).

% not_bit_Suc_0_numeral
thf(fact_5877_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_5878_of__nat__diff,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M2 @ N2 ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_5879_of__nat__diff,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N2 ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_5880_of__nat__diff,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N2 ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_5881_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 )
       => ( ! [M3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M3 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_5882_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N3: nat] :
          ( M2
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( M2
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% int_cases4
thf(fact_5883_int__zle__neg,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
      = ( ( N2 = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_5884_int__ops_I4_J,axiom,
    ! [A: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_5885_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_5886_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z6: int] :
        ? [N: nat] :
          ( Z6
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_5887_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_5888_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% nonpos_int_cases
thf(fact_5889_int__sum,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% int_sum
thf(fact_5890_int__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% int_sum
thf(fact_5891_mod__mult2__eq_H,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5892_mod__mult2__eq_H,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5893_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N3 )
          & ( K
            = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_5894_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N3: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% pos_int_cases
thf(fact_5895_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
       => ~ ! [N3: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% int_cases3
thf(fact_5896_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_5897_not__zle__0__negative,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).

% not_zle_0_negative
thf(fact_5898_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N3: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% negD
thf(fact_5899_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_5900_nat__approx__posE,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ~ ! [N3: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_5901_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_5902_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_5903_inverse__of__nat__le,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( N2 != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_5904_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% neg_int_cases
thf(fact_5905_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_5906_double__arith__series,axiom,
    ! [A: complex,D: complex,N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5907_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
        @ ( groups7108830773950497114d_enat
          @ ^ [I3: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5908_double__arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5909_double__arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5910_double__arith__series,axiom,
    ! [A: real,D: real,N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5911_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).

% double_gauss_sum
thf(fact_5912_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ) ) ).

% double_gauss_sum
thf(fact_5913_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_5914_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_5915_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).

% double_gauss_sum
thf(fact_5916_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_5917_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_5918_of__nat__code__if,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N: nat] :
          ( if_complex @ ( N = zero_zero_nat ) @ zero_zero_complex
          @ ( produc1917071388513777916omplex
            @ ^ [M: nat,Q5: nat] : ( if_complex @ ( Q5 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5919_of__nat__code__if,axiom,
    ( semiri4216267220026989637d_enat
    = ( ^ [N: nat] :
          ( if_Extended_enat @ ( N = zero_zero_nat ) @ zero_z5237406670263579293d_enat
          @ ( produc2676513652042109336d_enat
            @ ^ [M: nat,Q5: nat] : ( if_Extended_enat @ ( Q5 = zero_zero_nat ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M ) ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M ) ) @ one_on7984719198319812577d_enat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5920_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M: nat,Q5: nat] : ( if_real @ ( Q5 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5921_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M: nat,Q5: nat] : ( if_int @ ( Q5 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M ) ) @ one_one_int ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5922_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M: nat,Q5: nat] : ( if_nat @ ( Q5 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M ) ) @ one_one_nat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5923_lemma__termdiff2,axiom,
    ! [H2: complex,Z3: complex,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z3 @ H2 ) @ N2 ) @ ( power_power_complex @ Z3 @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H2
          @ ( groups2073611262835488442omplex
            @ ^ [P6: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z3 @ H2 ) @ Q5 ) @ ( power_power_complex @ Z3 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_5924_lemma__termdiff2,axiom,
    ! [H2: real,Z3: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z3 @ H2 ) @ N2 ) @ ( power_power_real @ Z3 @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H2
          @ ( groups6591440286371151544t_real
            @ ^ [P6: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z3 @ H2 ) @ Q5 ) @ ( power_power_real @ Z3 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_5925_lemma__termdiff3,axiom,
    ! [H2: real,Z3: real,K5: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z3 ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z3 @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z3 @ H2 ) @ N2 ) @ ( power_power_real @ Z3 @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_5926_lemma__termdiff3,axiom,
    ! [H2: complex,Z3: complex,K5: real,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z3 ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z3 @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z3 @ H2 ) @ N2 ) @ ( power_power_complex @ Z3 @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_5927_pochhammer__double,axiom,
    ! [Z3: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s2602460028002588243omplex @ Z3 @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z3 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_5928_pochhammer__double,axiom,
    ! [Z3: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s7457072308508201937r_real @ Z3 @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_5929_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I3: complex] : ( plus_plus_complex @ I3 @ one_one_complex )
          @ N
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_5930_of__nat__code,axiom,
    ( semiri4216267220026989637d_enat
    = ( ^ [N: nat] :
          ( semiri8563196900006977889d_enat
          @ ^ [I3: extended_enat] : ( plus_p3455044024723400733d_enat @ I3 @ one_on7984719198319812577d_enat )
          @ N
          @ zero_z5237406670263579293d_enat ) ) ) ).

% of_nat_code
thf(fact_5931_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I3: real] : ( plus_plus_real @ I3 @ one_one_real )
          @ N
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_5932_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
          @ N
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_5933_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
          @ N
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_5934_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B2 @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_5935_lessThan__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( set_ord_lessThan_nat @ X )
        = ( set_ord_lessThan_nat @ Y ) )
      = ( X = Y ) ) ).

% lessThan_eq_iff
thf(fact_5936_lessThan__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( set_ord_lessThan_int @ X )
        = ( set_ord_lessThan_int @ Y ) )
      = ( X = Y ) ) ).

% lessThan_eq_iff
thf(fact_5937_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
      = ( ord_less_set_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_5938_lessThan__iff,axiom,
    ! [I: extended_enat,K: extended_enat] :
      ( ( member_Extended_enat @ I @ ( set_or8419480210114673929d_enat @ K ) )
      = ( ord_le72135733267957522d_enat @ I @ K ) ) ).

% lessThan_iff
thf(fact_5939_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_5940_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_5941_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_5942_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_5943_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_5944_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_5945_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_5946_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_5947_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_5948_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_5949_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_5950_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_5951_single__Diff__lessThan,axiom,
    ! [K: extended_enat] :
      ( ( minus_925952699566721837d_enat @ ( insert_Extended_enat @ K @ bot_bo7653980558646680370d_enat ) @ ( set_or8419480210114673929d_enat @ K ) )
      = ( insert_Extended_enat @ K @ bot_bo7653980558646680370d_enat ) ) ).

% single_Diff_lessThan
thf(fact_5952_single__Diff__lessThan,axiom,
    ! [K: real] :
      ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
      = ( insert_real @ K @ bot_bot_set_real ) ) ).

% single_Diff_lessThan
thf(fact_5953_single__Diff__lessThan,axiom,
    ! [K: nat] :
      ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
      = ( insert_nat @ K @ bot_bot_set_nat ) ) ).

% single_Diff_lessThan
thf(fact_5954_single__Diff__lessThan,axiom,
    ! [K: int] :
      ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
      = ( insert_int @ K @ bot_bot_set_int ) ) ).

% single_Diff_lessThan
thf(fact_5955_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_5956_sum_OlessThan__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_5957_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_5958_sum_OlessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_5959_int__int__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M2 = N2 ) ) ).

% int_int_eq
thf(fact_5960_lessThan__non__empty,axiom,
    ! [X: real] :
      ( ( set_or5984915006950818249n_real @ X )
     != bot_bot_set_real ) ).

% lessThan_non_empty
thf(fact_5961_lessThan__non__empty,axiom,
    ! [X: int] :
      ( ( set_ord_lessThan_int @ X )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_5962_infinite__Iio,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).

% infinite_Iio
thf(fact_5963_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_set_nat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_5964_lessThan__def,axiom,
    ( set_or8419480210114673929d_enat
    = ( ^ [U2: extended_enat] :
          ( collec4429806609662206161d_enat
          @ ^ [X4: extended_enat] : ( ord_le72135733267957522d_enat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_5965_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X4: real] : ( ord_less_real @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_5966_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_5967_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_int @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_5968_Iio__eq__empty__iff,axiom,
    ! [N2: extended_enat] :
      ( ( ( set_or8419480210114673929d_enat @ N2 )
        = bot_bo7653980558646680370d_enat )
      = ( N2 = bot_bo4199563552545308370d_enat ) ) ).

% Iio_eq_empty_iff
thf(fact_5969_Iio__eq__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
thf(fact_5970_lessThan__strict__subset__iff,axiom,
    ! [M2: extended_enat,N2: extended_enat] :
      ( ( ord_le2529575680413868914d_enat @ ( set_or8419480210114673929d_enat @ M2 ) @ ( set_or8419480210114673929d_enat @ N2 ) )
      = ( ord_le72135733267957522d_enat @ M2 @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_5971_lessThan__strict__subset__iff,axiom,
    ! [M2: real,N2: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M2 ) @ ( set_or5984915006950818249n_real @ N2 ) )
      = ( ord_less_real @ M2 @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_5972_lessThan__strict__subset__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_5973_lessThan__strict__subset__iff,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N2 ) )
      = ( ord_less_int @ M2 @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_5974_pochhammer__pos,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_5975_pochhammer__pos,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_5976_pochhammer__pos,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_5977_pochhammer__eq__0__mono,axiom,
    ! [A: real,N2: nat,M2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( comm_s7457072308508201937r_real @ A @ M2 )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_5978_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N2: nat,M2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
        = zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( comm_s2602460028002588243omplex @ A @ M2 )
          = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_5979_pochhammer__neq__0__mono,axiom,
    ! [A: real,M2: nat,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M2 )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( comm_s7457072308508201937r_real @ A @ N2 )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_5980_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M2: nat,N2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ M2 )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( ( comm_s2602460028002588243omplex @ A @ N2 )
         != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_5981_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_5982_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_5983_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).

% finite_nat_bounded
thf(fact_5984_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K2: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_5985_pochhammer__nonneg,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_5986_pochhammer__nonneg,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_5987_pochhammer__nonneg,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_5988_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_5989_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
          = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_5990_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_5991_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
          = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
          = zero_zero_complex ) ) ) ).

% pochhammer_0_left
thf(fact_5992_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s3181272606743183617d_enat @ zero_z5237406670263579293d_enat @ N2 )
          = one_on7984719198319812577d_enat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s3181272606743183617d_enat @ zero_z5237406670263579293d_enat @ N2 )
          = zero_z5237406670263579293d_enat ) ) ) ).

% pochhammer_0_left
thf(fact_5993_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_5994_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_5995_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_5996_sum__diff__distrib,axiom,
    ! [Q: int > nat,P: int > nat,N2: int] :
      ( ! [X5: int] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N2 ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N2 ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( minus_minus_nat @ ( P @ X4 ) @ ( Q @ X4 ) )
          @ ( set_ord_lessThan_int @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_5997_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N2: nat] :
      ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( P @ X4 ) @ ( Q @ X4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_5998_log__of__power__le,axiom,
    ! [M2: nat,B2: real,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B2 @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_eq_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_le
thf(fact_5999_log__of__power__less,axiom,
    ! [M2: nat,B2: real,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B2 @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_less
thf(fact_6000_pochhammer__rec,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_6001_pochhammer__rec,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_6002_pochhammer__rec,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_6003_pochhammer__rec,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_6004_pochhammer__rec,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( comm_s3181272606743183617d_enat @ A @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ A @ ( comm_s3181272606743183617d_enat @ ( plus_p3455044024723400733d_enat @ A @ one_on7984719198319812577d_enat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_6005_pochhammer__rec_H,axiom,
    ! [Z3: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ Z3 @ ( suc @ N2 ) )
      = ( times_times_complex @ ( plus_plus_complex @ Z3 @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z3 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_6006_pochhammer__rec_H,axiom,
    ! [Z3: extended_enat,N2: nat] :
      ( ( comm_s3181272606743183617d_enat @ Z3 @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ Z3 @ ( semiri4216267220026989637d_enat @ N2 ) ) @ ( comm_s3181272606743183617d_enat @ Z3 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_6007_pochhammer__rec_H,axiom,
    ! [Z3: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z3 @ ( suc @ N2 ) )
      = ( times_times_real @ ( plus_plus_real @ Z3 @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z3 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_6008_pochhammer__rec_H,axiom,
    ! [Z3: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z3 @ ( suc @ N2 ) )
      = ( times_times_int @ ( plus_plus_int @ Z3 @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z3 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_6009_pochhammer__rec_H,axiom,
    ! [Z3: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z3 @ ( suc @ N2 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z3 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z3 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_6010_pochhammer__Suc,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N2 ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6011_pochhammer__Suc,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( comm_s3181272606743183617d_enat @ A @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( comm_s3181272606743183617d_enat @ A @ N2 ) @ ( plus_p3455044024723400733d_enat @ A @ ( semiri4216267220026989637d_enat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6012_pochhammer__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N2 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6013_pochhammer__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6014_pochhammer__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6015_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6016_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6017_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6018_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
        = zero_zero_complex )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6019_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
        = zero_zero_real )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6020_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
        = zero_zero_int )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6021_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
        = zero_zero_complex )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_6022_pochhammer__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_6023_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6024_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6025_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6026_pochhammer__product_H,axiom,
    ! [Z3: complex,N2: nat,M2: nat] :
      ( ( comm_s2602460028002588243omplex @ Z3 @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z3 @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z3 @ ( semiri8010041392384452111omplex @ N2 ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_6027_pochhammer__product_H,axiom,
    ! [Z3: extended_enat,N2: nat,M2: nat] :
      ( ( comm_s3181272606743183617d_enat @ Z3 @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( times_7803423173614009249d_enat @ ( comm_s3181272606743183617d_enat @ Z3 @ N2 ) @ ( comm_s3181272606743183617d_enat @ ( plus_p3455044024723400733d_enat @ Z3 @ ( semiri4216267220026989637d_enat @ N2 ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_6028_pochhammer__product_H,axiom,
    ! [Z3: real,N2: nat,M2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z3 @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z3 @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z3 @ ( semiri5074537144036343181t_real @ N2 ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_6029_pochhammer__product_H,axiom,
    ! [Z3: int,N2: nat,M2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z3 @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z3 @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z3 @ ( semiri1314217659103216013at_int @ N2 ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_6030_pochhammer__product_H,axiom,
    ! [Z3: nat,N2: nat,M2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z3 @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z3 @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z3 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_6031_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6032_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( G @ zero_zero_nat )
        @ ( groups7108830773950497114d_enat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6033_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6034_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6035_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_int @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6036_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_real @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6037_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6038_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6039_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_6040_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_6041_le__log2__of__power,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% le_log2_of_power
thf(fact_6042_one__diff__power__eq,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_6043_one__diff__power__eq,axiom,
    ! [X: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_6044_one__diff__power__eq,axiom,
    ! [X: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_6045_power__diff__1__eq,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_6046_power__diff__1__eq,axiom,
    ! [X: complex,N2: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_6047_power__diff__1__eq,axiom,
    ! [X: real,N2: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_6048_geometric__sum,axiom,
    ! [X: complex,N2: nat] :
      ( ( X != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_6049_geometric__sum,axiom,
    ! [X: real,N2: nat] :
      ( ( X != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_6050_pochhammer__product,axiom,
    ! [M2: nat,N2: nat,Z3: complex] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( comm_s2602460028002588243omplex @ Z3 @ N2 )
        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z3 @ M2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z3 @ ( semiri8010041392384452111omplex @ M2 ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_6051_pochhammer__product,axiom,
    ! [M2: nat,N2: nat,Z3: extended_enat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( comm_s3181272606743183617d_enat @ Z3 @ N2 )
        = ( times_7803423173614009249d_enat @ ( comm_s3181272606743183617d_enat @ Z3 @ M2 ) @ ( comm_s3181272606743183617d_enat @ ( plus_p3455044024723400733d_enat @ Z3 @ ( semiri4216267220026989637d_enat @ M2 ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_6052_pochhammer__product,axiom,
    ! [M2: nat,N2: nat,Z3: real] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( comm_s7457072308508201937r_real @ Z3 @ N2 )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z3 @ M2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z3 @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_6053_pochhammer__product,axiom,
    ! [M2: nat,N2: nat,Z3: int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( comm_s4660882817536571857er_int @ Z3 @ N2 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z3 @ M2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z3 @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_6054_pochhammer__product,axiom,
    ! [M2: nat,N2: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( comm_s4663373288045622133er_nat @ Z3 @ N2 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z3 @ M2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z3 @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_6055_less__log2__of__power,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M2 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% less_log2_of_power
thf(fact_6056_sum__gp__strict,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri8010041392384452111omplex @ N2 ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6057_sum__gp__strict,axiom,
    ! [X: real,N2: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ N2 ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6058_lemma__termdiff1,axiom,
    ! [Z3: int,H2: int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P6: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z3 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_int @ Z3 @ P6 ) ) @ ( power_power_int @ Z3 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ Z3 @ P6 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z3 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_int @ Z3 @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6059_lemma__termdiff1,axiom,
    ! [Z3: complex,H2: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P6: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z3 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_complex @ Z3 @ P6 ) ) @ ( power_power_complex @ Z3 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ Z3 @ P6 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z3 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_complex @ Z3 @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6060_lemma__termdiff1,axiom,
    ! [Z3: real,H2: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P6: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z3 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_real @ Z3 @ P6 ) ) @ ( power_power_real @ Z3 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ Z3 @ P6 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z3 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_real @ Z3 @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6061_diff__power__eq__sum,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N2 ) ) @ ( power_power_int @ Y @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ X @ P6 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6062_diff__power__eq__sum,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) @ ( power_power_complex @ Y @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ X @ P6 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6063_diff__power__eq__sum,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N2 ) ) @ ( power_power_real @ Y @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ X @ P6 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6064_power__diff__sumr2,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6065_power__diff__sumr2,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6066_power__diff__sumr2,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6067_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_6068_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_6069_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_6070_log2__of__power__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_less
thf(fact_6071_log2__of__power__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_le
thf(fact_6072_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > int,K5: int,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6073_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > nat,K5: nat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6074_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > real,K5: real,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6075_one__diff__power__eq_H,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6076_one__diff__power__eq_H,axiom,
    ! [X: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6077_one__diff__power__eq_H,axiom,
    ! [X: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6078_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_split_even_odd
thf(fact_6079_pochhammer__minus,axiom,
    ! [B2: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6080_pochhammer__minus,axiom,
    ! [B2: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6081_pochhammer__minus,axiom,
    ! [B2: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6082_pochhammer__minus_H,axiom,
    ! [B2: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6083_pochhammer__minus_H,axiom,
    ! [B2: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6084_pochhammer__minus_H,axiom,
    ! [B2: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6085_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B2 @ N2 ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_6086_ceiling__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_6087_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_6088_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_6089_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_6090_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_6091_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( ( real_V7735802525324610683m_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_6092_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( ( real_V1022390504157884413omplex @ X )
        = zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_6093_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_6094_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_6095_nonzero__norm__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_6096_nonzero__norm__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_6097_norm__diff__ineq,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) ) ).

% norm_diff_ineq
thf(fact_6098_norm__diff__ineq,axiom,
    ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) ) ).

% norm_diff_ineq
thf(fact_6099_norm__uminus__minus,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_6100_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_6101_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N2: nat,Z3: real] :
      ( ( ( power_power_real @ W2 @ N2 )
        = ( power_power_real @ Z3 @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V7735802525324610683m_real @ W2 )
          = ( real_V7735802525324610683m_real @ Z3 ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_6102_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N2: nat,Z3: complex] :
      ( ( ( power_power_complex @ W2 @ N2 )
        = ( power_power_complex @ Z3 @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V1022390504157884413omplex @ W2 )
          = ( real_V1022390504157884413omplex @ Z3 ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_6103_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_6104_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_6105_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_6106_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_6107_norm__add__leD,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_6108_norm__add__leD,axiom,
    ! [A: complex,B2: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_6109_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_6110_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_6111_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_6112_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_6113_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B2: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_6114_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B2: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_6115_power__eq__1__iff,axiom,
    ! [W2: real,N2: nat] :
      ( ( ( power_power_real @ W2 @ N2 )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W2 )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_6116_power__eq__1__iff,axiom,
    ! [W2: complex,N2: nat] :
      ( ( ( power_power_complex @ W2 @ N2 )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W2 )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_6117_norm__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_6118_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B2: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_6119_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_6120_sum__bounds__lt__plus1,axiom,
    ! [F: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_6121_sumr__cos__zero__one,axiom,
    ! [N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M: nat] : ( times_times_real @ ( cos_coeff @ M ) @ ( power_power_real @ zero_zero_real @ M ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_6122_pochhammer__times__pochhammer__half,axiom,
    ! [Z3: complex,N2: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z3 @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z3 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K2: nat] : ( plus_plus_complex @ Z3 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_6123_pochhammer__times__pochhammer__half,axiom,
    ! [Z3: real,N2: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z3 @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K2: nat] : ( plus_plus_real @ Z3 @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_6124_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A3: complex,N: nat] :
          ( if_complex @ ( N = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_6125_pochhammer__code,axiom,
    ( comm_s3181272606743183617d_enat
    = ( ^ [A3: extended_enat,N: nat] :
          ( if_Extended_enat @ ( N = zero_zero_nat ) @ one_on7984719198319812577d_enat
          @ ( set_fo2538466533108834004d_enat
            @ ^ [O: nat] : ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A3 @ ( semiri4216267220026989637d_enat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_on7984719198319812577d_enat ) ) ) ) ).

% pochhammer_code
thf(fact_6126_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A3: real,N: nat] :
          ( if_real @ ( N = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_6127_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A3: int,N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_6128_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A3: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_6129_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B2 @ X ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_6130_geometric__deriv__sums,axiom,
    ! [Z3: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z3 ) @ one_one_real )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( power_power_real @ Z3 @ N ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_6131_geometric__deriv__sums,axiom,
    ! [Z3: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z3 ) @ one_one_real )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( power_power_complex @ Z3 @ N ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_6132_powr__eq__0__iff,axiom,
    ! [W2: real,Z3: real] :
      ( ( ( powr_real @ W2 @ Z3 )
        = zero_zero_real )
      = ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
thf(fact_6133_powr__0,axiom,
    ! [Z3: real] :
      ( ( powr_real @ zero_zero_real @ Z3 )
      = zero_zero_real ) ).

% powr_0
thf(fact_6134_of__nat__prod,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_6135_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [X4: nat] : ( semiri5074537144036343181t_real @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_6136_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_6137_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( semiri1316708129612266289at_nat @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_6138_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [X4: nat] : ( ring_1_of_int_real @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_6139_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( ring_1_of_int_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_6140_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups2316167850115554303t_real
        @ ^ [X4: int] : ( ring_1_of_int_real @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_6141_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( ring_1_of_int_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_6142_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups861055069439313189ex_nat @ F @ A2 )
          = zero_zero_nat )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6143_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
          = zero_zero_nat )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6144_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2880970938130013265at_nat @ F @ A2 )
          = zero_zero_nat )
        = ( ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
              & ( ( F @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6145_prod__zero__iff,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups129246275422532515t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( ( F @ X4 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6146_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups766887009212190081x_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ( F @ X4 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6147_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups2316167850115554303t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ( F @ X4 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6148_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups97031904164794029t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
              & ( ( F @ X4 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6149_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups858564598930262913ex_int @ F @ A2 )
          = zero_zero_int )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ( F @ X4 )
                = zero_zero_int ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6150_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2878480467620962989at_int @ F @ A2 )
          = zero_zero_int )
        = ( ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
              & ( ( F @ X4 )
                = zero_zero_int ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6151_prod__zero__iff,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups6464643781859351333omplex @ F @ A2 )
          = zero_zero_complex )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( ( F @ X4 )
                = zero_zero_complex ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6152_prod_Oempty,axiom,
    ! [G: extended_enat > nat] :
      ( ( groups2880970938130013265at_nat @ G @ bot_bo7653980558646680370d_enat )
      = one_one_nat ) ).

% prod.empty
thf(fact_6153_prod_Oempty,axiom,
    ! [G: extended_enat > int] :
      ( ( groups2878480467620962989at_int @ G @ bot_bo7653980558646680370d_enat )
      = one_one_int ) ).

% prod.empty
thf(fact_6154_prod_Oempty,axiom,
    ! [G: extended_enat > complex] :
      ( ( groups4622424608036095791omplex @ G @ bot_bo7653980558646680370d_enat )
      = one_one_complex ) ).

% prod.empty
thf(fact_6155_prod_Oempty,axiom,
    ! [G: extended_enat > real] :
      ( ( groups97031904164794029t_real @ G @ bot_bo7653980558646680370d_enat )
      = one_one_real ) ).

% prod.empty
thf(fact_6156_prod_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
      = one_one_nat ) ).

% prod.empty
thf(fact_6157_prod_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
      = one_one_int ) ).

% prod.empty
thf(fact_6158_prod_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
      = one_one_complex ) ).

% prod.empty
thf(fact_6159_prod_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_6160_prod_Oempty,axiom,
    ! [G: nat > complex] :
      ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
      = one_one_complex ) ).

% prod.empty
thf(fact_6161_prod_Oempty,axiom,
    ! [G: nat > real] :
      ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
      = one_one_real ) ).

% prod.empty
thf(fact_6162_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups861055069439313189ex_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_6163_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_6164_prod_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2880970938130013265at_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_6165_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups858564598930262913ex_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.infinite
thf(fact_6166_prod_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > int] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2878480467620962989at_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.infinite
thf(fact_6167_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups6464643781859351333omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_6168_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_6169_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups7440179247065528705omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_6170_prod_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > complex] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4622424608036095791omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_6171_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_6172_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_6173_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ( dvd_dvd_nat @ ( F @ A ) @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_6174_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ A @ A2 )
       => ( dvd_dvd_nat @ ( F @ A ) @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_6175_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ( dvd_dvd_nat @ ( F @ A ) @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_6176_dvd__prodI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ A @ A2 )
       => ( dvd_dvd_nat @ ( F @ A ) @ ( groups2880970938130013265at_nat @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_6177_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ( dvd_dvd_int @ ( F @ A ) @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_6178_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ A @ A2 )
       => ( dvd_dvd_int @ ( F @ A ) @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_6179_dvd__prodI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ A @ A2 )
       => ( dvd_dvd_int @ ( F @ A ) @ ( groups2878480467620962989at_int @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_6180_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( dvd_dvd_int @ ( F @ A ) @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_6181_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ( dvd_dvd_int @ ( F @ A ) @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_6182_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( dvd_dvd_nat @ ( F @ A ) @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_6183_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B2: nat,F: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ( ( B2
            = ( F @ A ) )
         => ( dvd_dvd_nat @ B2 @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_6184_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B2: nat,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ A @ A2 )
       => ( ( B2
            = ( F @ A ) )
         => ( dvd_dvd_nat @ B2 @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_6185_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B2: nat,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ( ( B2
            = ( F @ A ) )
         => ( dvd_dvd_nat @ B2 @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_6186_dvd__prod__eqI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B2: nat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ A @ A2 )
       => ( ( B2
            = ( F @ A ) )
         => ( dvd_dvd_nat @ B2 @ ( groups2880970938130013265at_nat @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_6187_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B2: int,F: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ( ( B2
            = ( F @ A ) )
         => ( dvd_dvd_int @ B2 @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_6188_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B2: int,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ A @ A2 )
       => ( ( B2
            = ( F @ A ) )
         => ( dvd_dvd_int @ B2 @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_6189_dvd__prod__eqI,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,B2: int,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ A @ A2 )
       => ( ( B2
            = ( F @ A ) )
         => ( dvd_dvd_int @ B2 @ ( groups2878480467620962989at_int @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_6190_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B2: int,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( ( B2
            = ( F @ A ) )
         => ( dvd_dvd_int @ B2 @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_6191_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B2: int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ( ( B2
            = ( F @ A ) )
         => ( dvd_dvd_int @ B2 @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_6192_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B2: nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( ( B2
            = ( F @ A ) )
         => ( dvd_dvd_nat @ B2 @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_6193_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_6194_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B2: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups4696554848551431203al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups4696554848551431203al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = one_one_nat ) ) ) ) ).

% prod.delta
thf(fact_6195_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B2: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups861055069439313189ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups861055069439313189ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = one_one_nat ) ) ) ) ).

% prod.delta
thf(fact_6196_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B2: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = one_one_nat ) ) ) ) ).

% prod.delta
thf(fact_6197_prod_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2880970938130013265at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2880970938130013265at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = one_one_nat ) ) ) ) ).

% prod.delta
thf(fact_6198_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B2: real > int] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups4694064378042380927al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups4694064378042380927al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = one_one_int ) ) ) ) ).

% prod.delta
thf(fact_6199_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B2: complex > int] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups858564598930262913ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups858564598930262913ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = one_one_int ) ) ) ) ).

% prod.delta
thf(fact_6200_prod_Odelta,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2878480467620962989at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2878480467620962989at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = one_one_int ) ) ) ) ).

% prod.delta
thf(fact_6201_prod_Odelta,axiom,
    ! [S2: set_real,A: real,B2: real > complex] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups713298508707869441omplex
              @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups713298508707869441omplex
              @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_6202_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B2: nat > complex] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_6203_prod_Odelta,axiom,
    ! [S2: set_complex,A: complex,B2: complex > complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_6204_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups4696554848551431203al_nat
              @ ^ [K2: real] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups4696554848551431203al_nat
              @ ^ [K2: real] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = one_one_nat ) ) ) ) ).

% prod.delta'
thf(fact_6205_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B2: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups861055069439313189ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups861055069439313189ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = one_one_nat ) ) ) ) ).

% prod.delta'
thf(fact_6206_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B2: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [K2: int] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [K2: int] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = one_one_nat ) ) ) ) ).

% prod.delta'
thf(fact_6207_prod_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2880970938130013265at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2880970938130013265at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_nat )
              @ S2 )
            = one_one_nat ) ) ) ) ).

% prod.delta'
thf(fact_6208_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: real > int] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups4694064378042380927al_int
              @ ^ [K2: real] : ( if_int @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups4694064378042380927al_int
              @ ^ [K2: real] : ( if_int @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = one_one_int ) ) ) ) ).

% prod.delta'
thf(fact_6209_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B2: complex > int] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups858564598930262913ex_int
              @ ^ [K2: complex] : ( if_int @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups858564598930262913ex_int
              @ ^ [K2: complex] : ( if_int @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = one_one_int ) ) ) ) ).

% prod.delta'
thf(fact_6210_prod_Odelta_H,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2878480467620962989at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2878480467620962989at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_int )
              @ S2 )
            = one_one_int ) ) ) ) ).

% prod.delta'
thf(fact_6211_prod_Odelta_H,axiom,
    ! [S2: set_real,A: real,B2: real > complex] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups713298508707869441omplex
              @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups713298508707869441omplex
              @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_6212_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B2: nat > complex] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_6213_prod_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B2: complex > complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = ( B2 @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B2 @ K2 ) @ one_one_complex )
              @ S2 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_6214_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups4696554848551431203al_nat @ G @ ( insert_real @ X @ A2 ) )
          = ( times_times_nat @ ( G @ X ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6215_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups861055069439313189ex_nat @ G @ ( insert_complex @ X @ A2 ) )
          = ( times_times_nat @ ( G @ X ) @ ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6216_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups1707563613775114915nt_nat @ G @ ( insert_int @ X @ A2 ) )
          = ( times_times_nat @ ( G @ X ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6217_prod_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ~ ( member_Extended_enat @ X @ A2 )
       => ( ( groups2880970938130013265at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
          = ( times_times_nat @ ( G @ X ) @ ( groups2880970938130013265at_nat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6218_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups4694064378042380927al_int @ G @ ( insert_real @ X @ A2 ) )
          = ( times_times_int @ ( G @ X ) @ ( groups4694064378042380927al_int @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6219_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups858564598930262913ex_int @ G @ ( insert_complex @ X @ A2 ) )
          = ( times_times_int @ ( G @ X ) @ ( groups858564598930262913ex_int @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6220_prod_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ~ ( member_Extended_enat @ X @ A2 )
       => ( ( groups2878480467620962989at_int @ G @ ( insert_Extended_enat @ X @ A2 ) )
          = ( times_times_int @ ( G @ X ) @ ( groups2878480467620962989at_int @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6221_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6222_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6223_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6224_prod_OlessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6225_prod_OlessThan__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6226_prod_OlessThan__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6227_prod_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6228_prod_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6229_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_6230_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X: complex] :
      ( ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_6231_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6232_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6233_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > extended_enat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = one_on7984719198319812577d_enat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6234_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6235_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6236_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_nat,G: real > nat > int,R: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups4694064378042380927al_int
            @ ^ [X4: real] :
                ( groups705719431365010083at_int @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y5: nat] :
                ( groups4694064378042380927al_int
                @ ^ [X4: real] : ( G @ X4 @ Y5 )
                @ ( collect_real
                  @ ^ [X4: real] :
                      ( ( member_real @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% prod.swap_restrict
thf(fact_6237_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B: set_nat,G: complex > nat > int,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups858564598930262913ex_int
            @ ^ [X4: complex] :
                ( groups705719431365010083at_int @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y5: nat] :
                ( groups858564598930262913ex_int
                @ ^ [X4: complex] : ( G @ X4 @ Y5 )
                @ ( collect_complex
                  @ ^ [X4: complex] :
                      ( ( member_complex @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% prod.swap_restrict
thf(fact_6238_prod_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B: set_nat,G: extended_enat > nat > int,R: extended_enat > nat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups2878480467620962989at_int
            @ ^ [X4: extended_enat] :
                ( groups705719431365010083at_int @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y5: nat] :
                ( groups2878480467620962989at_int
                @ ^ [X4: extended_enat] : ( G @ X4 @ Y5 )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X4: extended_enat] :
                      ( ( member_Extended_enat @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% prod.swap_restrict
thf(fact_6239_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_int,G: real > int > int,R: real > int > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_int @ B )
       => ( ( groups4694064378042380927al_int
            @ ^ [X4: real] :
                ( groups1705073143266064639nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y5: int] :
                      ( ( member_int @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y5: int] :
                ( groups4694064378042380927al_int
                @ ^ [X4: real] : ( G @ X4 @ Y5 )
                @ ( collect_real
                  @ ^ [X4: real] :
                      ( ( member_real @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% prod.swap_restrict
thf(fact_6240_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B: set_int,G: complex > int > int,R: complex > int > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_int @ B )
       => ( ( groups858564598930262913ex_int
            @ ^ [X4: complex] :
                ( groups1705073143266064639nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y5: int] :
                      ( ( member_int @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y5: int] :
                ( groups858564598930262913ex_int
                @ ^ [X4: complex] : ( G @ X4 @ Y5 )
                @ ( collect_complex
                  @ ^ [X4: complex] :
                      ( ( member_complex @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% prod.swap_restrict
thf(fact_6241_prod_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B: set_int,G: extended_enat > int > int,R: extended_enat > int > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite_finite_int @ B )
       => ( ( groups2878480467620962989at_int
            @ ^ [X4: extended_enat] :
                ( groups1705073143266064639nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y5: int] :
                      ( ( member_int @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y5: int] :
                ( groups2878480467620962989at_int
                @ ^ [X4: extended_enat] : ( G @ X4 @ Y5 )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X4: extended_enat] :
                      ( ( member_Extended_enat @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% prod.swap_restrict
thf(fact_6242_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B: set_nat,G: real > nat > nat,R: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups4696554848551431203al_nat
            @ ^ [X4: real] :
                ( groups708209901874060359at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y5: nat] :
                ( groups4696554848551431203al_nat
                @ ^ [X4: real] : ( G @ X4 @ Y5 )
                @ ( collect_real
                  @ ^ [X4: real] :
                      ( ( member_real @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% prod.swap_restrict
thf(fact_6243_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups861055069439313189ex_nat
            @ ^ [X4: complex] :
                ( groups708209901874060359at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y5: nat] :
                ( groups861055069439313189ex_nat
                @ ^ [X4: complex] : ( G @ X4 @ Y5 )
                @ ( collect_complex
                  @ ^ [X4: complex] :
                      ( ( member_complex @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% prod.swap_restrict
thf(fact_6244_prod_Oswap__restrict,axiom,
    ! [A2: set_int,B: set_nat,G: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups1707563613775114915nt_nat
            @ ^ [X4: int] :
                ( groups708209901874060359at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y5: nat] :
                ( groups1707563613775114915nt_nat
                @ ^ [X4: int] : ( G @ X4 @ Y5 )
                @ ( collect_int
                  @ ^ [X4: int] :
                      ( ( member_int @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% prod.swap_restrict
thf(fact_6245_prod_Oswap__restrict,axiom,
    ! [A2: set_Extended_enat,B: set_nat,G: extended_enat > nat > nat,R: extended_enat > nat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( groups2880970938130013265at_nat
            @ ^ [X4: extended_enat] :
                ( groups708209901874060359at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y5: nat] :
                      ( ( member_nat @ Y5 @ B )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y5: nat] :
                ( groups2880970938130013265at_nat
                @ ^ [X4: extended_enat] : ( G @ X4 @ Y5 )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X4: extended_enat] :
                      ( ( member_Extended_enat @ X4 @ A2 )
                      & ( R @ X4 @ Y5 ) ) ) )
            @ B ) ) ) ) ).

% prod.swap_restrict
thf(fact_6246_prod__mono,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
      ( ! [I4: extended_enat] :
          ( ( member_Extended_enat @ I4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
            & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A2 ) @ ( groups97031904164794029t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6247_prod__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
            & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6248_prod__mono,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
            & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6249_prod__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
            & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6250_prod__mono,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ! [I4: extended_enat] :
          ( ( member_Extended_enat @ I4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
            & ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) @ ( groups2880970938130013265at_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6251_prod__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
            & ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6252_prod__mono,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
            & ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6253_prod__mono,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > int,G: extended_enat > int] :
      ( ! [I4: extended_enat] :
          ( ( member_Extended_enat @ I4 @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
            & ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_int @ ( groups2878480467620962989at_int @ F @ A2 ) @ ( groups2878480467620962989at_int @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6254_prod__mono,axiom,
    ! [A2: set_real,F: real > int,G: real > int] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
            & ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6255_prod__mono,axiom,
    ! [A2: set_nat,F: nat > int,G: nat > int] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
            & ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6256_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_6257_prod__nonneg,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_6258_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_6259_prod__pos,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_6260_prod__pos,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_6261_prod__pos,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_6262_prod__ge__1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups97031904164794029t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6263_prod__ge__1,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6264_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6265_prod__ge__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6266_prod__ge__1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6267_prod__ge__1,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6268_prod__ge__1,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6269_prod__ge__1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > int] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ord_less_eq_int @ one_one_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ one_one_int @ ( groups2878480467620962989at_int @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6270_prod__ge__1,axiom,
    ! [A2: set_real,F: real > int] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_int @ one_one_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6271_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_eq_int @ one_one_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ one_one_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6272_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_6273_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_6274_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ? [X2: extended_enat] :
            ( ( member_Extended_enat @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_nat ) )
       => ( ( groups2880970938130013265at_nat @ F @ A2 )
          = zero_zero_nat ) ) ) ).

% prod_zero
thf(fact_6275_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_6276_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_6277_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_6278_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_6279_prod__zero,axiom,
    ! [A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ? [X2: complex] :
            ( ( member_complex @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_int ) )
       => ( ( groups858564598930262913ex_int @ F @ A2 )
          = zero_zero_int ) ) ) ).

% prod_zero
thf(fact_6280_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ? [X2: extended_enat] :
            ( ( member_Extended_enat @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_int ) )
       => ( ( groups2878480467620962989at_int @ F @ A2 )
          = zero_zero_int ) ) ) ).

% prod_zero
thf(fact_6281_prod__zero,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A2 )
            & ( ( F @ X2 )
              = zero_zero_complex ) )
       => ( ( groups6464643781859351333omplex @ F @ A2 )
          = zero_zero_complex ) ) ) ).

% prod_zero
thf(fact_6282_prod__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B2: nat] :
      ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A3: nat] : ( times_times_real @ ( F @ A3 ) )
        @ A
        @ B2
        @ one_one_real ) ) ).

% prod_atLeastAtMost_code
thf(fact_6283_prod__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B2: nat] :
      ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A3: nat] : ( times_times_complex @ ( F @ A3 ) )
        @ A
        @ B2
        @ one_one_complex ) ) ).

% prod_atLeastAtMost_code
thf(fact_6284_prod__atLeastAtMost__code,axiom,
    ! [F: nat > extended_enat,A: nat,B2: nat] :
      ( ( groups7961826882256487087d_enat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2538466533108834004d_enat
        @ ^ [A3: nat] : ( times_7803423173614009249d_enat @ ( F @ A3 ) )
        @ A
        @ B2
        @ one_on7984719198319812577d_enat ) ) ).

% prod_atLeastAtMost_code
thf(fact_6285_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B2: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A3: nat] : ( times_times_int @ ( F @ A3 ) )
        @ A
        @ B2
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_6286_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B2: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A3: nat] : ( times_times_nat @ ( F @ A3 ) )
        @ A
        @ B2
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_6287_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > nat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4696554848551431203al_nat @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups4696554848551431203al_nat
          @ ^ [X4: real] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_nat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6288_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > nat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups861055069439313189ex_nat @ G
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups861055069439313189ex_nat
          @ ^ [X4: complex] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_nat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6289_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > nat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups1707563613775114915nt_nat
          @ ^ [X4: int] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_nat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6290_prod_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2880970938130013265at_nat @ G
          @ ( collec4429806609662206161d_enat
            @ ^ [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups2880970938130013265at_nat
          @ ^ [X4: extended_enat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_nat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6291_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > int,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4694064378042380927al_int @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups4694064378042380927al_int
          @ ^ [X4: real] : ( if_int @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_int )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6292_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > int,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups858564598930262913ex_int @ G
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups858564598930262913ex_int
          @ ^ [X4: complex] : ( if_int @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_int )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6293_prod_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > int,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2878480467620962989at_int @ G
          @ ( collec4429806609662206161d_enat
            @ ^ [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups2878480467620962989at_int
          @ ^ [X4: extended_enat] : ( if_int @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_int )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6294_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups713298508707869441omplex @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups713298508707869441omplex
          @ ^ [X4: real] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6295_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups6464643781859351333omplex @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups6464643781859351333omplex
          @ ^ [X4: nat] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6296_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > complex,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A2 )
                & ( P @ X4 ) ) ) )
        = ( groups3708469109370488835omplex
          @ ^ [X4: complex] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6297_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_6298_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_6299_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M2: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_6300_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M2: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_6301_prod__le__1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) )
            & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ one_on7984719198319812577d_enat ) ) )
     => ( ord_le2932123472753598470d_enat @ ( groups8932437906259616549d_enat @ F @ A2 ) @ one_on7984719198319812577d_enat ) ) ).

% prod_le_1
thf(fact_6302_prod__le__1,axiom,
    ! [A2: set_real,F: real > extended_enat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) )
            & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ one_on7984719198319812577d_enat ) ) )
     => ( ord_le2932123472753598470d_enat @ ( groups7973222482632965587d_enat @ F @ A2 ) @ one_on7984719198319812577d_enat ) ) ).

% prod_le_1
thf(fact_6303_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > extended_enat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) )
            & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ one_on7984719198319812577d_enat ) ) )
     => ( ord_le2932123472753598470d_enat @ ( groups7961826882256487087d_enat @ F @ A2 ) @ one_on7984719198319812577d_enat ) ) ).

% prod_le_1
thf(fact_6304_prod__le__1,axiom,
    ! [A2: set_int,F: int > extended_enat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) )
            & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ one_on7984719198319812577d_enat ) ) )
     => ( ord_le2932123472753598470d_enat @ ( groups5078248829458667347d_enat @ F @ A2 ) @ one_on7984719198319812577d_enat ) ) ).

% prod_le_1
thf(fact_6305_prod__le__1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6306_prod__le__1,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6307_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6308_prod__le__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6309_prod__le__1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ! [X5: extended_enat] :
          ( ( member_Extended_enat @ X5 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
            & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_6310_prod__le__1,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
            & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_6311_prod_Orelated,axiom,
    ! [R: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R @ one_one_nat @ one_one_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( times_times_nat @ X1 @ Y1 ) @ ( times_times_nat @ X23 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups861055069439313189ex_nat @ H2 @ S2 ) @ ( groups861055069439313189ex_nat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_6312_prod_Orelated,axiom,
    ! [R: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
      ( ( R @ one_one_nat @ one_one_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( times_times_nat @ X1 @ Y1 ) @ ( times_times_nat @ X23 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_6313_prod_Orelated,axiom,
    ! [R: nat > nat > $o,S2: set_Extended_enat,H2: extended_enat > nat,G: extended_enat > nat] :
      ( ( R @ one_one_nat @ one_one_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( times_times_nat @ X1 @ Y1 ) @ ( times_times_nat @ X23 @ Y22 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2880970938130013265at_nat @ H2 @ S2 ) @ ( groups2880970938130013265at_nat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_6314_prod_Orelated,axiom,
    ! [R: int > int > $o,S2: set_complex,H2: complex > int,G: complex > int] :
      ( ( R @ one_one_int @ one_one_int )
     => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( times_times_int @ X1 @ Y1 ) @ ( times_times_int @ X23 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups858564598930262913ex_int @ H2 @ S2 ) @ ( groups858564598930262913ex_int @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_6315_prod_Orelated,axiom,
    ! [R: int > int > $o,S2: set_Extended_enat,H2: extended_enat > int,G: extended_enat > int] :
      ( ( R @ one_one_int @ one_one_int )
     => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( times_times_int @ X1 @ Y1 ) @ ( times_times_int @ X23 @ Y22 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2878480467620962989at_int @ H2 @ S2 ) @ ( groups2878480467620962989at_int @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_6316_prod_Orelated,axiom,
    ! [R: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_6317_prod_Orelated,axiom,
    ! [R: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_6318_prod_Orelated,axiom,
    ! [R: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_6319_prod_Orelated,axiom,
    ! [R: real > real > $o,S2: set_Extended_enat,H2: extended_enat > real,G: extended_enat > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y22 ) ) )
       => ( ( finite4001608067531595151d_enat @ S2 )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups97031904164794029t_real @ H2 @ S2 ) @ ( groups97031904164794029t_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_6320_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups6464643781859351333omplex @ H2 @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_6321_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups4696554848551431203al_nat @ G @ ( insert_real @ X @ A2 ) )
            = ( groups4696554848551431203al_nat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups4696554848551431203al_nat @ G @ ( insert_real @ X @ A2 ) )
            = ( times_times_nat @ ( G @ X ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6322_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups861055069439313189ex_nat @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups861055069439313189ex_nat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups861055069439313189ex_nat @ G @ ( insert_complex @ X @ A2 ) )
            = ( times_times_nat @ ( G @ X ) @ ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6323_prod_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups1707563613775114915nt_nat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups1707563613775114915nt_nat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups1707563613775114915nt_nat @ G @ ( insert_int @ X @ A2 ) )
            = ( times_times_nat @ ( G @ X ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6324_prod_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ X @ A2 )
         => ( ( groups2880970938130013265at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
            = ( groups2880970938130013265at_nat @ G @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ X @ A2 )
         => ( ( groups2880970938130013265at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
            = ( times_times_nat @ ( G @ X ) @ ( groups2880970938130013265at_nat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6325_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups4694064378042380927al_int @ G @ ( insert_real @ X @ A2 ) )
            = ( groups4694064378042380927al_int @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups4694064378042380927al_int @ G @ ( insert_real @ X @ A2 ) )
            = ( times_times_int @ ( G @ X ) @ ( groups4694064378042380927al_int @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6326_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups858564598930262913ex_int @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups858564598930262913ex_int @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups858564598930262913ex_int @ G @ ( insert_complex @ X @ A2 ) )
            = ( times_times_int @ ( G @ X ) @ ( groups858564598930262913ex_int @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6327_prod_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ X @ A2 )
         => ( ( groups2878480467620962989at_int @ G @ ( insert_Extended_enat @ X @ A2 ) )
            = ( groups2878480467620962989at_int @ G @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ X @ A2 )
         => ( ( groups2878480467620962989at_int @ G @ ( insert_Extended_enat @ X @ A2 ) )
            = ( times_times_int @ ( G @ X ) @ ( groups2878480467620962989at_int @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6328_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1681761925125756287l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6329_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups129246275422532515t_real @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
            = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6330_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups766887009212190081x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6331_prod__dvd__prod__subset,axiom,
    ! [B: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6332_prod__dvd__prod__subset,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( dvd_dvd_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) @ ( groups2880970938130013265at_nat @ F @ B ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6333_prod__dvd__prod__subset,axiom,
    ! [B: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6334_prod__dvd__prod__subset,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( dvd_dvd_int @ ( groups2878480467620962989at_int @ F @ A2 ) @ ( groups2878480467620962989at_int @ F @ B ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6335_prod__dvd__prod__subset,axiom,
    ! [B: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B )
     => ( ( ord_less_eq_set_int @ A2 @ B )
       => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ F @ B ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6336_prod__dvd__prod__subset,axiom,
    ! [B: set_nat,A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6337_prod__dvd__prod__subset,axiom,
    ! [B: set_int,A2: set_int,F: int > int] :
      ( ( finite_finite_int @ B )
     => ( ( ord_less_eq_set_int @ A2 @ B )
       => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6338_prod__dvd__prod__subset,axiom,
    ! [B: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6339_prod__dvd__prod__subset2,axiom,
    ! [B: set_real,A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6340_prod__dvd__prod__subset2,axiom,
    ! [B: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6341_prod__dvd__prod__subset2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( ! [A4: extended_enat] :
              ( ( member_Extended_enat @ A4 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) @ ( groups2880970938130013265at_nat @ G @ B ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6342_prod__dvd__prod__subset2,axiom,
    ! [B: set_real,A2: set_real,F: real > int,G: real > int] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ A2 )
             => ( dvd_dvd_int @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6343_prod__dvd__prod__subset2,axiom,
    ! [B: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ A2 )
             => ( dvd_dvd_int @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6344_prod__dvd__prod__subset2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > int,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( ! [A4: extended_enat] :
              ( ( member_Extended_enat @ A4 @ A2 )
             => ( dvd_dvd_int @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_int @ ( groups2878480467620962989at_int @ F @ A2 ) @ ( groups2878480467620962989at_int @ G @ B ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6345_prod__dvd__prod__subset2,axiom,
    ! [B: set_int,A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ B )
     => ( ( ord_less_eq_set_int @ A2 @ B )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6346_prod__dvd__prod__subset2,axiom,
    ! [B: set_nat,A2: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ! [A4: nat] :
              ( ( member_nat @ A4 @ A2 )
             => ( dvd_dvd_int @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ B ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6347_prod__dvd__prod__subset2,axiom,
    ! [B: set_int,A2: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ B )
     => ( ( ord_less_eq_set_int @ A2 @ B )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ A2 )
             => ( dvd_dvd_int @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ G @ B ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6348_prod__dvd__prod__subset2,axiom,
    ! [B: set_nat,A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ! [A4: nat] :
              ( ( member_nat @ A4 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ G @ B ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6349_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = one_one_nat ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = one_one_nat ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups4696554848551431203al_nat @ G @ S2 )
                        = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6350_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > nat,H2: complex > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
               => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = one_one_nat ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = one_one_nat ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups4696554848551431203al_nat @ G @ S2 )
                        = ( groups861055069439313189ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6351_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > nat,H2: int > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
               => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = one_one_nat ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = one_one_nat ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups4696554848551431203al_nat @ G @ S2 )
                        = ( groups1707563613775114915nt_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6352_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_Extended_enat,S2: set_real,I: extended_enat > real,J: real > extended_enat,T3: set_Extended_enat,G: real > nat,H2: extended_enat > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite4001608067531595151d_enat @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S5 ) )
               => ( member_Extended_enat @ ( J @ A4 ) @ ( minus_925952699566721837d_enat @ T3 @ T5 ) ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: extended_enat] :
                    ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = one_one_nat ) )
                 => ( ! [B4: extended_enat] :
                        ( ( member_Extended_enat @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = one_one_nat ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups4696554848551431203al_nat @ G @ S2 )
                        = ( groups2880970938130013265at_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6353_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_real,S2: set_complex,I: real > complex,J: complex > real,T3: set_real,G: complex > nat,H2: real > nat] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = one_one_nat ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = one_one_nat ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups861055069439313189ex_nat @ G @ S2 )
                        = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6354_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_complex,S2: set_complex,I: complex > complex,J: complex > complex,T3: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = one_one_nat ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = one_one_nat ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups861055069439313189ex_nat @ G @ S2 )
                        = ( groups861055069439313189ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6355_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,G: complex > nat,H2: int > nat] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = one_one_nat ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = one_one_nat ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups861055069439313189ex_nat @ G @ S2 )
                        = ( groups1707563613775114915nt_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6356_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T5: set_Extended_enat,S2: set_complex,I: extended_enat > complex,J: complex > extended_enat,T3: set_Extended_enat,G: complex > nat,H2: extended_enat > nat] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite4001608067531595151d_enat @ T5 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
               => ( member_Extended_enat @ ( J @ A4 ) @ ( minus_925952699566721837d_enat @ T3 @ T5 ) ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: extended_enat] :
                    ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = one_one_nat ) )
                 => ( ! [B4: extended_enat] :
                        ( ( member_Extended_enat @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = one_one_nat ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups861055069439313189ex_nat @ G @ S2 )
                        = ( groups2880970938130013265at_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6357_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > nat,H2: real > nat] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = one_one_nat ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = one_one_nat ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups1707563613775114915nt_nat @ G @ S2 )
                        = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6358_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,G: int > nat,H2: complex > nat] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S2 @ S5 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S2 @ S5 ) )
               => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S5 )
                     => ( ( G @ A4 )
                        = one_one_nat ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = one_one_nat ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S2 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups1707563613775114915nt_nat @ G @ S2 )
                        = ( groups861055069439313189ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6359_powr__add,axiom,
    ! [X: real,A: real,B2: real] :
      ( ( powr_real @ X @ ( plus_plus_real @ A @ B2 ) )
      = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B2 ) ) ) ).

% powr_add
thf(fact_6360_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4696554848551431203al_nat @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = one_one_nat ) ) ) )
        = ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_6361_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups861055069439313189ex_nat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = one_one_nat ) ) ) )
        = ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_6362_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = one_one_nat ) ) ) )
        = ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_6363_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2880970938130013265at_nat @ G
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X4: extended_enat] :
                  ( ( G @ X4 )
                  = one_one_nat ) ) ) )
        = ( groups2880970938130013265at_nat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_6364_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4694064378042380927al_int @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = one_one_int ) ) ) )
        = ( groups4694064378042380927al_int @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_6365_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups858564598930262913ex_int @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = one_one_int ) ) ) )
        = ( groups858564598930262913ex_int @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_6366_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2878480467620962989at_int @ G
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X4: extended_enat] :
                  ( ( G @ X4 )
                  = one_one_int ) ) ) )
        = ( groups2878480467620962989at_int @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_6367_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > complex] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups713298508707869441omplex @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = one_one_complex ) ) ) )
        = ( groups713298508707869441omplex @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_6368_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = one_one_complex ) ) ) )
        = ( groups3708469109370488835omplex @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_6369_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups7440179247065528705omplex @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = one_one_complex ) ) ) )
        = ( groups7440179247065528705omplex @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_6370_prod_Onat__diff__reindex,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_6371_prod_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_6372_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N2: nat,M2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_6373_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_6374_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6375_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I4: nat] :
                ( ( member_nat @ I4 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6376_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I4: complex] :
                ( ( member_complex @ I4 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6377_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I4: int] :
                ( ( member_int @ I4 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6378_less__1__prod2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( member_Extended_enat @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups97031904164794029t_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6379_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: real > int] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6380_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I4: complex] :
                ( ( member_complex @ I4 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6381_less__1__prod2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( member_Extended_enat @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups2878480467620962989at_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6382_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: nat > int] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I4: nat] :
                ( ( member_nat @ I4 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups705719431365010083at_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6383_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: int > int] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I4: int] :
                ( ( member_int @ I4 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups1705073143266064639nt_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6384_less__1__prod,axiom,
    ! [I6: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6385_less__1__prod,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups97031904164794029t_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6386_less__1__prod,axiom,
    ! [I6: set_real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6387_less__1__prod,axiom,
    ! [I6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6388_less__1__prod,axiom,
    ! [I6: set_int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6389_less__1__prod,axiom,
    ! [I6: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6390_less__1__prod,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups2878480467620962989at_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6391_less__1__prod,axiom,
    ! [I6: set_real,F: real > int] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6392_less__1__prod,axiom,
    ! [I6: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups705719431365010083at_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6393_less__1__prod,axiom,
    ! [I6: set_int,F: int > int] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups1705073143266064639nt_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6394_prod_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups861055069439313189ex_nat @ G @ A2 )
          = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B ) ) @ ( groups861055069439313189ex_nat @ G @ B ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6395_prod_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > nat] :
      ( ( ord_le7203529160286727270d_enat @ B @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2880970938130013265at_nat @ G @ A2 )
          = ( times_times_nat @ ( groups2880970938130013265at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ B ) ) @ ( groups2880970938130013265at_nat @ G @ B ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6396_prod_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups858564598930262913ex_int @ G @ A2 )
          = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B ) ) @ ( groups858564598930262913ex_int @ G @ B ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6397_prod_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > int] :
      ( ( ord_le7203529160286727270d_enat @ B @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2878480467620962989at_int @ G @ A2 )
          = ( times_times_int @ ( groups2878480467620962989at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ B ) ) @ ( groups2878480467620962989at_int @ G @ B ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6398_prod_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups766887009212190081x_real @ G @ A2 )
          = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B ) ) @ ( groups766887009212190081x_real @ G @ B ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6399_prod_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > real] :
      ( ( ord_le7203529160286727270d_enat @ B @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups97031904164794029t_real @ G @ A2 )
          = ( times_times_real @ ( groups97031904164794029t_real @ G @ ( minus_925952699566721837d_enat @ A2 @ B ) ) @ ( groups97031904164794029t_real @ G @ B ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6400_prod_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: complex > complex] :
      ( ( ord_le211207098394363844omplex @ B @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups3708469109370488835omplex @ G @ A2 )
          = ( times_times_complex @ ( groups3708469109370488835omplex @ G @ ( minus_811609699411566653omplex @ A2 @ B ) ) @ ( groups3708469109370488835omplex @ G @ B ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6401_prod_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > complex] :
      ( ( ord_le7203529160286727270d_enat @ B @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups4622424608036095791omplex @ G @ A2 )
          = ( times_times_complex @ ( groups4622424608036095791omplex @ G @ ( minus_925952699566721837d_enat @ A2 @ B ) ) @ ( groups4622424608036095791omplex @ G @ B ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6402_prod_Osubset__diff,axiom,
    ! [B: set_complex,A2: set_complex,G: complex > extended_enat] :
      ( ( ord_le211207098394363844omplex @ B @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups8780218893797010257d_enat @ G @ A2 )
          = ( times_7803423173614009249d_enat @ ( groups8780218893797010257d_enat @ G @ ( minus_811609699411566653omplex @ A2 @ B ) ) @ ( groups8780218893797010257d_enat @ G @ B ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6403_prod_Osubset__diff,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ B @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups8932437906259616549d_enat @ G @ A2 )
          = ( times_7803423173614009249d_enat @ ( groups8932437906259616549d_enat @ G @ ( minus_925952699566721837d_enat @ A2 @ B ) ) @ ( groups8932437906259616549d_enat @ G @ B ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6404_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups4696554848551431203al_nat @ G @ A2 )
                  = ( groups4696554848551431203al_nat @ H2 @ B ) )
                = ( ( groups4696554848551431203al_nat @ G @ C4 )
                  = ( groups4696554848551431203al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6405_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_nat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups861055069439313189ex_nat @ G @ A2 )
                  = ( groups861055069439313189ex_nat @ H2 @ B ) )
                = ( ( groups861055069439313189ex_nat @ G @ C4 )
                  = ( groups861055069439313189ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6406_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_nat ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups2880970938130013265at_nat @ G @ A2 )
                  = ( groups2880970938130013265at_nat @ H2 @ B ) )
                = ( ( groups2880970938130013265at_nat @ G @ C4 )
                  = ( groups2880970938130013265at_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6407_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_int ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups4694064378042380927al_int @ G @ A2 )
                  = ( groups4694064378042380927al_int @ H2 @ B ) )
                = ( ( groups4694064378042380927al_int @ G @ C4 )
                  = ( groups4694064378042380927al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6408_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_int ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups858564598930262913ex_int @ G @ A2 )
                  = ( groups858564598930262913ex_int @ H2 @ B ) )
                = ( ( groups858564598930262913ex_int @ G @ C4 )
                  = ( groups858564598930262913ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6409_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > int,H2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_int ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups2878480467620962989at_int @ G @ A2 )
                  = ( groups2878480467620962989at_int @ H2 @ B ) )
                = ( ( groups2878480467620962989at_int @ G @ C4 )
                  = ( groups2878480467620962989at_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6410_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ A2 )
                  = ( groups713298508707869441omplex @ H2 @ B ) )
                = ( ( groups713298508707869441omplex @ G @ C4 )
                  = ( groups713298508707869441omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6411_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ A2 )
                  = ( groups3708469109370488835omplex @ H2 @ B ) )
                = ( ( groups3708469109370488835omplex @ G @ C4 )
                  = ( groups3708469109370488835omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6412_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > complex,H2: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups4622424608036095791omplex @ G @ A2 )
                  = ( groups4622424608036095791omplex @ H2 @ B ) )
                = ( ( groups4622424608036095791omplex @ G @ C4 )
                  = ( groups4622424608036095791omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6413_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_real ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ A2 )
                  = ( groups1681761925125756287l_real @ H2 @ B ) )
                = ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6414_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups4696554848551431203al_nat @ G @ C4 )
                  = ( groups4696554848551431203al_nat @ H2 @ C4 ) )
               => ( ( groups4696554848551431203al_nat @ G @ A2 )
                  = ( groups4696554848551431203al_nat @ H2 @ B ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6415_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_nat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups861055069439313189ex_nat @ G @ C4 )
                  = ( groups861055069439313189ex_nat @ H2 @ C4 ) )
               => ( ( groups861055069439313189ex_nat @ G @ A2 )
                  = ( groups861055069439313189ex_nat @ H2 @ B ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6416_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_nat ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups2880970938130013265at_nat @ G @ C4 )
                  = ( groups2880970938130013265at_nat @ H2 @ C4 ) )
               => ( ( groups2880970938130013265at_nat @ G @ A2 )
                  = ( groups2880970938130013265at_nat @ H2 @ B ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6417_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_int ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups4694064378042380927al_int @ G @ C4 )
                  = ( groups4694064378042380927al_int @ H2 @ C4 ) )
               => ( ( groups4694064378042380927al_int @ G @ A2 )
                  = ( groups4694064378042380927al_int @ H2 @ B ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6418_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_int ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups858564598930262913ex_int @ G @ C4 )
                  = ( groups858564598930262913ex_int @ H2 @ C4 ) )
               => ( ( groups858564598930262913ex_int @ G @ A2 )
                  = ( groups858564598930262913ex_int @ H2 @ B ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6419_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > int,H2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_int ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups2878480467620962989at_int @ G @ C4 )
                  = ( groups2878480467620962989at_int @ H2 @ C4 ) )
               => ( ( groups2878480467620962989at_int @ G @ A2 )
                  = ( groups2878480467620962989at_int @ H2 @ B ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6420_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ C4 )
                  = ( groups713298508707869441omplex @ H2 @ C4 ) )
               => ( ( groups713298508707869441omplex @ G @ A2 )
                  = ( groups713298508707869441omplex @ H2 @ B ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6421_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ C4 )
                  = ( groups3708469109370488835omplex @ H2 @ C4 ) )
               => ( ( groups3708469109370488835omplex @ G @ A2 )
                  = ( groups3708469109370488835omplex @ H2 @ B ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6422_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B: set_Extended_enat,G: extended_enat > complex,H2: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups4622424608036095791omplex @ G @ C4 )
                  = ( groups4622424608036095791omplex @ H2 @ C4 ) )
               => ( ( groups4622424608036095791omplex @ G @ A2 )
                  = ( groups4622424608036095791omplex @ H2 @ B ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6423_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = one_one_real ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) )
               => ( ( groups1681761925125756287l_real @ G @ A2 )
                  = ( groups1681761925125756287l_real @ H2 @ B ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6424_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ S2 )
            = ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6425_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups2880970938130013265at_nat @ G @ S2 )
            = ( groups2880970938130013265at_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6426_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ S2 )
            = ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6427_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups2878480467620962989at_int @ G @ S2 )
            = ( groups2878480467620962989at_int @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6428_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ S2 )
            = ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6429_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups4622424608036095791omplex @ G @ S2 )
            = ( groups4622424608036095791omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6430_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ S2 )
            = ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6431_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups97031904164794029t_real @ G @ S2 )
            = ( groups97031904164794029t_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6432_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G @ S2 )
            = ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6433_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G @ S2 )
            = ( groups129246275422532515t_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6434_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ T3 )
            = ( groups861055069439313189ex_nat @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6435_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups2880970938130013265at_nat @ G @ T3 )
            = ( groups2880970938130013265at_nat @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6436_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ T3 )
            = ( groups858564598930262913ex_int @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6437_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups2878480467620962989at_int @ G @ T3 )
            = ( groups2878480467620962989at_int @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6438_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ T3 )
            = ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6439_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups4622424608036095791omplex @ G @ T3 )
            = ( groups4622424608036095791omplex @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6440_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ T3 )
            = ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6441_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups97031904164794029t_real @ G @ T3 )
            = ( groups97031904164794029t_real @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6442_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G @ T3 )
            = ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6443_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G @ T3 )
            = ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6444_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4696554848551431203al_nat @ G @ S2 )
              = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6445_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_nat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups861055069439313189ex_nat @ G @ S2 )
              = ( groups861055069439313189ex_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6446_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H2: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_nat ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2880970938130013265at_nat @ G @ S2 )
              = ( groups2880970938130013265at_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6447_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_int ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4694064378042380927al_int @ G @ S2 )
              = ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6448_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_int ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups858564598930262913ex_int @ G @ S2 )
              = ( groups858564598930262913ex_int @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6449_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H2: extended_enat > int,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_int ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2878480467620962989at_int @ G @ S2 )
              = ( groups2878480467620962989at_int @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6450_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups713298508707869441omplex @ G @ S2 )
              = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6451_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_complex ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ S2 )
              = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6452_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,H2: extended_enat > complex,G: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_complex ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4622424608036095791omplex @ G @ S2 )
              = ( groups4622424608036095791omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6453_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ S2 )
              = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6454_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4696554848551431203al_nat @ G @ T3 )
              = ( groups4696554848551431203al_nat @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6455_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups861055069439313189ex_nat @ G @ T3 )
              = ( groups861055069439313189ex_nat @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6456_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2880970938130013265at_nat @ G @ T3 )
              = ( groups2880970938130013265at_nat @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6457_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4694064378042380927al_int @ G @ T3 )
              = ( groups4694064378042380927al_int @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6458_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups858564598930262913ex_int @ G @ T3 )
              = ( groups858564598930262913ex_int @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6459_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > int,H2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2878480467620962989at_int @ G @ T3 )
              = ( groups2878480467620962989at_int @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6460_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups713298508707869441omplex @ G @ T3 )
              = ( groups713298508707869441omplex @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6461_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ T3 )
              = ( groups3708469109370488835omplex @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6462_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S2: set_Extended_enat,G: extended_enat > complex,H2: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S2 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4622424608036095791omplex @ G @ T3 )
              = ( groups4622424608036095791omplex @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6463_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ T3 )
              = ( groups1681761925125756287l_real @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6464_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6465_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6466_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6467_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6468_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6469_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_6470_powser__sums__zero,axiom,
    ! [A: nat > complex] :
      ( sums_complex
      @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_6471_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_real @ ( G @ M2 ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6472_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_complex @ ( G @ M2 ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6473_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_7803423173614009249d_enat @ ( G @ M2 ) @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6474_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_int @ ( G @ M2 ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6475_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_nat @ ( G @ M2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6476_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( times_times_real @ ( G @ ( suc @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6477_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( times_times_complex @ ( G @ ( suc @ N2 ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6478_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( times_7803423173614009249d_enat @ ( G @ ( suc @ N2 ) ) @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6479_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ ( suc @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6480_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6481_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6482_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( G @ zero_zero_nat )
        @ ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6483_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( G @ zero_zero_nat )
        @ ( groups7961826882256487087d_enat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6484_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6485_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6486_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_real @ ( G @ M2 )
          @ ( groups129246275422532515t_real
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6487_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_complex @ ( G @ M2 )
          @ ( groups6464643781859351333omplex
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6488_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_7803423173614009249d_enat @ ( G @ M2 )
          @ ( groups7961826882256487087d_enat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6489_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ M2 )
          @ ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6490_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ M2 )
          @ ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6491_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_6492_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_6493_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
              & ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6494_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [I4: extended_enat] :
            ( ( member_Extended_enat @ I4 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
              & ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => ( ord_less_real @ ( groups97031904164794029t_real @ F @ A2 ) @ ( groups97031904164794029t_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6495_prod__mono__strict,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
              & ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_real )
         => ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6496_prod__mono__strict,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
              & ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_nat )
         => ( ord_less_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6497_prod__mono__strict,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
              & ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_int )
         => ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6498_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ A2 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
              & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6499_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [I4: extended_enat] :
            ( ( member_Extended_enat @ I4 @ A2 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
              & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => ( ord_less_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) @ ( groups2880970938130013265at_nat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6500_prod__mono__strict,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ A2 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
              & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_real )
         => ( ord_less_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6501_prod__mono__strict,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ A2 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
              & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_int )
         => ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6502_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ A2 )
           => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
              & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6503_even__prod__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A2 ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_6504_even__prod__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_6505_even__prod__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups2880970938130013265at_nat @ F @ A2 ) )
        = ( ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_6506_even__prod__iff,axiom,
    ! [A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A2 ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_6507_even__prod__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups2878480467620962989at_int @ F @ A2 ) )
        = ( ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_6508_even__prod__iff,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A2 ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_6509_even__prod__iff,axiom,
    ! [A2: set_int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A2 ) )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_6510_even__prod__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A2 ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_6511_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups861055069439313189ex_nat @ G @ ( insert_complex @ X @ A2 ) )
        = ( times_times_nat @ ( G @ X ) @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6512_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2880970938130013265at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( times_times_nat @ ( G @ X ) @ ( groups2880970938130013265at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6513_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > nat,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4696554848551431203al_nat @ G @ ( insert_real @ X @ A2 ) )
        = ( times_times_nat @ ( G @ X ) @ ( groups4696554848551431203al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6514_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > nat,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G @ ( insert_int @ X @ A2 ) )
        = ( times_times_nat @ ( G @ X ) @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6515_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > int,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups858564598930262913ex_int @ G @ ( insert_complex @ X @ A2 ) )
        = ( times_times_int @ ( G @ X ) @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6516_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > int,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2878480467620962989at_int @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( times_times_int @ ( G @ X ) @ ( groups2878480467620962989at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6517_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > int,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4694064378042380927al_int @ G @ ( insert_real @ X @ A2 ) )
        = ( times_times_int @ ( G @ X ) @ ( groups4694064378042380927al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6518_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
        = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6519_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > real,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups97031904164794029t_real @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( times_times_real @ ( G @ X ) @ ( groups97031904164794029t_real @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6520_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
        = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6521_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups861055069439313189ex_nat @ G @ A2 )
          = ( times_times_nat @ ( G @ X ) @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6522_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2880970938130013265at_nat @ G @ A2 )
          = ( times_times_nat @ ( G @ X ) @ ( groups2880970938130013265at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6523_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups4696554848551431203al_nat @ G @ A2 )
          = ( times_times_nat @ ( G @ X ) @ ( groups4696554848551431203al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6524_prod_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups1707563613775114915nt_nat @ G @ A2 )
          = ( times_times_nat @ ( G @ X ) @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6525_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups858564598930262913ex_int @ G @ A2 )
          = ( times_times_int @ ( G @ X ) @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6526_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2878480467620962989at_int @ G @ A2 )
          = ( times_times_int @ ( G @ X ) @ ( groups2878480467620962989at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6527_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups4694064378042380927al_int @ G @ A2 )
          = ( times_times_int @ ( G @ X ) @ ( groups4694064378042380927al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6528_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups766887009212190081x_real @ G @ A2 )
          = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6529_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups97031904164794029t_real @ G @ A2 )
          = ( times_times_real @ ( G @ X ) @ ( groups97031904164794029t_real @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6530_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups1681761925125756287l_real @ G @ A2 )
          = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6531_prod_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > real,P5: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P5 ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P5 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6532_prod_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > complex,P5: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P5 ) ) )
        = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P5 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6533_prod_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > extended_enat,P5: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P5 ) ) )
        = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P5 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6534_prod_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > int,P5: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P5 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P5 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6535_prod_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > nat,P5: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P5 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P5 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6536_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups861055069439313189ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( times_times_nat @ ( B2 @ A ) @ ( groups861055069439313189ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups861055069439313189ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups861055069439313189ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6537_prod_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > nat,C: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2880970938130013265at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( times_times_nat @ ( B2 @ A ) @ ( groups2880970938130013265at_nat @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2880970938130013265at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups2880970938130013265at_nat @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6538_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B2: real > nat,C: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups4696554848551431203al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( times_times_nat @ ( B2 @ A ) @ ( groups4696554848551431203al_nat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups4696554848551431203al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups4696554848551431203al_nat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6539_prod_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B2: int > nat,C: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( times_times_nat @ ( B2 @ A ) @ ( groups1707563613775114915nt_nat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups1707563613775114915nt_nat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6540_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups858564598930262913ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( times_times_int @ ( B2 @ A ) @ ( groups858564598930262913ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups858564598930262913ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups858564598930262913ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6541_prod_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > int,C: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups2878480467620962989at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( times_times_int @ ( B2 @ A ) @ ( groups2878480467620962989at_int @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups2878480467620962989at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups2878480467620962989at_int @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6542_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B2: real > int,C: real > int] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups4694064378042380927al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( times_times_int @ ( B2 @ A ) @ ( groups4694064378042380927al_int @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups4694064378042380927al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups4694064378042380927al_int @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6543_prod_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B2: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( times_times_real @ ( B2 @ A ) @ ( groups766887009212190081x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups766887009212190081x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups766887009212190081x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6544_prod_Odelta__remove,axiom,
    ! [S2: set_Extended_enat,A: extended_enat,B2: extended_enat > real,C: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( ( member_Extended_enat @ A @ S2 )
         => ( ( groups97031904164794029t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( times_times_real @ ( B2 @ A ) @ ( groups97031904164794029t_real @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S2 )
         => ( ( groups97031904164794029t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups97031904164794029t_real @ C @ ( minus_925952699566721837d_enat @ S2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6545_prod_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B2: real > real,C: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( times_times_real @ ( B2 @ A ) @ ( groups1681761925125756287l_real @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups1681761925125756287l_real @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6546_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb @ Xa2 )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa2 )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_6547_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F5: nat > nat > nat,A3: nat,B3: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B3 @ A3 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F5 @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B3 @ ( F5 @ A3 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_6548_prod__mono2,axiom,
    ! [B: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A4 ) ) )
           => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ F @ B ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6549_prod__mono2,axiom,
    ! [B: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A4 ) ) )
           => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ F @ B ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6550_prod__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A4 ) ) )
           => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A2 ) @ ( groups97031904164794029t_real @ F @ B ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6551_prod__mono2,axiom,
    ! [B: set_real,A2: set_real,F: real > int] :
      ( ( finite_finite_real @ B )
     => ( ( ord_less_eq_set_real @ A2 @ B )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
           => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ F @ B ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6552_prod__mono2,axiom,
    ! [B: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A2 @ B )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
           => ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6553_prod__mono2,axiom,
    ! [B: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ B )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
           => ( ord_less_eq_int @ ( groups2878480467620962989at_int @ F @ A2 ) @ ( groups2878480467620962989at_int @ F @ B ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6554_prod__mono2,axiom,
    ! [B: set_nat,A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ! [B4: nat] :
              ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A4: nat] :
                ( ( member_nat @ A4 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A4 ) ) )
           => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ F @ B ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6555_prod__mono2,axiom,
    ! [B: set_int,A2: set_int,F: int > real] :
      ( ( finite_finite_int @ B )
     => ( ( ord_less_eq_set_int @ A2 @ B )
       => ( ! [B4: int] :
              ( ( member_int @ B4 @ ( minus_minus_set_int @ B @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A4 ) ) )
           => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ F @ B ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6556_prod__mono2,axiom,
    ! [B: set_nat,A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ! [B4: nat] :
              ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A4: nat] :
                ( ( member_nat @ A4 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
           => ( ord_less_eq_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6557_prod__mono2,axiom,
    ! [B: set_int,A2: set_int,F: int > int] :
      ( ( finite_finite_int @ B )
     => ( ( ord_less_eq_set_int @ A2 @ B )
       => ( ! [B4: int] :
              ( ( member_int @ B4 @ ( minus_minus_set_int @ B @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
           => ( ord_less_eq_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6558_prod__diff1,axiom,
    ! [A2: set_complex,F: complex > complex,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_complex @ A @ A2 )
           => ( ( groups3708469109370488835omplex @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide1717551699836669952omplex @ ( groups3708469109370488835omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A2 )
           => ( ( groups3708469109370488835omplex @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups3708469109370488835omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6559_prod__diff1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > complex,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_Extended_enat @ A @ A2 )
           => ( ( groups4622424608036095791omplex @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( divide1717551699836669952omplex @ ( groups4622424608036095791omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_Extended_enat @ A @ A2 )
           => ( ( groups4622424608036095791omplex @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( groups4622424608036095791omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6560_prod__diff1,axiom,
    ! [A2: set_real,F: real > complex,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_real @ A @ A2 )
           => ( ( groups713298508707869441omplex @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( divide1717551699836669952omplex @ ( groups713298508707869441omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_real @ A @ A2 )
           => ( ( groups713298508707869441omplex @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( groups713298508707869441omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6561_prod__diff1,axiom,
    ! [A2: set_int,F: int > complex,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_int @ A @ A2 )
           => ( ( groups7440179247065528705omplex @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( divide1717551699836669952omplex @ ( groups7440179247065528705omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_int @ A @ A2 )
           => ( ( groups7440179247065528705omplex @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( groups7440179247065528705omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6562_prod__diff1,axiom,
    ! [A2: set_nat,F: nat > complex,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_nat @ A @ A2 )
           => ( ( groups6464643781859351333omplex @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( divide1717551699836669952omplex @ ( groups6464643781859351333omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_nat @ A @ A2 )
           => ( ( groups6464643781859351333omplex @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( groups6464643781859351333omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6563_prod__diff1,axiom,
    ! [A2: set_complex,F: complex > nat,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( F @ A )
         != zero_zero_nat )
       => ( ( ( member_complex @ A @ A2 )
           => ( ( groups861055069439313189ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide_divide_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A2 )
           => ( ( groups861055069439313189ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6564_prod__diff1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( F @ A )
         != zero_zero_nat )
       => ( ( ( member_Extended_enat @ A @ A2 )
           => ( ( groups2880970938130013265at_nat @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( divide_divide_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_Extended_enat @ A @ A2 )
           => ( ( groups2880970938130013265at_nat @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( groups2880970938130013265at_nat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6565_prod__diff1,axiom,
    ! [A2: set_real,F: real > nat,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( F @ A )
         != zero_zero_nat )
       => ( ( ( member_real @ A @ A2 )
           => ( ( groups4696554848551431203al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( divide_divide_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_real @ A @ A2 )
           => ( ( groups4696554848551431203al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6566_prod__diff1,axiom,
    ! [A2: set_int,F: int > nat,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( F @ A )
         != zero_zero_nat )
       => ( ( ( member_int @ A @ A2 )
           => ( ( groups1707563613775114915nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( divide_divide_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_int @ A @ A2 )
           => ( ( groups1707563613775114915nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6567_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_6568_pochhammer__Suc__prod,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( comm_s3181272606743183617d_enat @ A @ ( suc @ N2 ) )
      = ( groups7961826882256487087d_enat
        @ ^ [I3: nat] : ( plus_p3455044024723400733d_enat @ A @ ( semiri4216267220026989637d_enat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6569_pochhammer__Suc__prod,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6570_pochhammer__Suc__prod,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6571_pochhammer__Suc__prod,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6572_pochhammer__prod__rev,axiom,
    ( comm_s3181272606743183617d_enat
    = ( ^ [A3: extended_enat,N: nat] :
          ( groups7961826882256487087d_enat
          @ ^ [I3: nat] : ( plus_p3455044024723400733d_enat @ A3 @ ( semiri4216267220026989637d_enat @ ( minus_minus_nat @ N @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6573_pochhammer__prod__rev,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A3: real,N: nat] :
          ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6574_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A3: int,N: nat] :
          ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6575_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A3: nat,N: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6576_prod_Oin__pairs,axiom,
    ! [G: nat > real,M2: nat,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_6577_prod_Oin__pairs,axiom,
    ! [G: nat > complex,M2: nat,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [I3: nat] : ( times_times_complex @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_6578_prod_Oin__pairs,axiom,
    ! [G: nat > extended_enat,M2: nat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups7961826882256487087d_enat
        @ ^ [I3: nat] : ( times_7803423173614009249d_enat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_6579_prod_Oin__pairs,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_6580_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_6581_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B2: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A3: nat] : ( plus_plus_int @ ( F @ A3 ) )
        @ A
        @ B2
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_6582_sum__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B2: nat] :
      ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A3: nat] : ( plus_plus_complex @ ( F @ A3 ) )
        @ A
        @ B2
        @ zero_zero_complex ) ) ).

% sum_atLeastAtMost_code
thf(fact_6583_sum__atLeastAtMost__code,axiom,
    ! [F: nat > extended_enat,A: nat,B2: nat] :
      ( ( groups7108830773950497114d_enat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2538466533108834004d_enat
        @ ^ [A3: nat] : ( plus_p3455044024723400733d_enat @ ( F @ A3 ) )
        @ A
        @ B2
        @ zero_z5237406670263579293d_enat ) ) ).

% sum_atLeastAtMost_code
thf(fact_6584_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B2: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A3: nat] : ( plus_plus_nat @ ( F @ A3 ) )
        @ A
        @ B2
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_6585_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B2: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A3: nat] : ( plus_plus_real @ ( F @ A3 ) )
        @ A
        @ B2
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_6586_pochhammer__Suc__prod__rev,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( comm_s3181272606743183617d_enat @ A @ ( suc @ N2 ) )
      = ( groups7961826882256487087d_enat
        @ ^ [I3: nat] : ( plus_p3455044024723400733d_enat @ A @ ( semiri4216267220026989637d_enat @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6587_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6588_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6589_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6590_power__half__series,axiom,
    ( sums_real
    @ ^ [N: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_6591_sums__zero,axiom,
    ( sums_nat
    @ ^ [N: nat] : zero_zero_nat
    @ zero_zero_nat ) ).

% sums_zero
thf(fact_6592_sums__zero,axiom,
    ( sums_real
    @ ^ [N: nat] : zero_zero_real
    @ zero_zero_real ) ).

% sums_zero
thf(fact_6593_sums__zero,axiom,
    ( sums_int
    @ ^ [N: nat] : zero_zero_int
    @ zero_zero_int ) ).

% sums_zero
thf(fact_6594_sums__zero,axiom,
    ( sums_complex
    @ ^ [N: nat] : zero_zero_complex
    @ zero_zero_complex ) ).

% sums_zero
thf(fact_6595_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S2: real,A2: set_nat,S5: real,F: nat > real] :
      ( ( sums_real @ G @ S2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( S5
            = ( plus_plus_real @ S2
              @ ( groups6591440286371151544t_real
                @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
                @ A2 ) ) )
         => ( sums_real
            @ ^ [N: nat] : ( if_real @ ( member_nat @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
            @ S5 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_6596_sums__iff__shift_H,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      = ( sums_real @ F @ S ) ) ).

% sums_iff_shift'
thf(fact_6597_sums__split__initial__segment,axiom,
    ! [F: nat > real,S: real,N2: nat] :
      ( ( sums_real @ F @ S )
     => ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_6598_sums__iff__shift,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_iff_shift
thf(fact_6599_prod__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups861055069439313189ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
             => ( ( F @ X4 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_6600_prod__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ( ( F @ X4 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_6601_prod__eq__1__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2880970938130013265at_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
             => ( ( F @ X4 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_6602_prod__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups708209901874060359at_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ( ( F @ X4 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_6603_prod__pos__nat__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6604_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 ) )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6605_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 ) )
        = ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6606_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 ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6607_int__prod,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% int_prod
thf(fact_6608_int__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A2 ) ) ).

% int_prod
thf(fact_6609_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : X4
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_6610_ln__prod,axiom,
    ! [I6: set_real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups1681761925125756287l_real @ F @ I6 ) )
          = ( groups8097168146408367636l_real
            @ ^ [X4: real] : ( ln_ln_real @ ( F @ X4 ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_6611_ln__prod,axiom,
    ! [I6: set_set_nat,F: set_nat > real] :
      ( ( finite1152437895449049373et_nat @ I6 )
     => ( ! [I4: set_nat] :
            ( ( member_set_nat @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups3619160379726066777t_real @ F @ I6 ) )
          = ( groups5107569545109728110t_real
            @ ^ [X4: set_nat] : ( ln_ln_real @ ( F @ X4 ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_6612_ln__prod,axiom,
    ! [I6: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups766887009212190081x_real @ F @ I6 ) )
          = ( groups5808333547571424918x_real
            @ ^ [X4: complex] : ( ln_ln_real @ ( F @ X4 ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_6613_ln__prod,axiom,
    ! [I6: set_int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups2316167850115554303t_real @ F @ I6 ) )
          = ( groups8778361861064173332t_real
            @ ^ [X4: int] : ( ln_ln_real @ ( F @ X4 ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_6614_ln__prod,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ! [I4: extended_enat] :
            ( ( member_Extended_enat @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups97031904164794029t_real @ F @ I6 ) )
          = ( groups4148127829035722712t_real
            @ ^ [X4: extended_enat] : ( ln_ln_real @ ( F @ X4 ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_6615_ln__prod,axiom,
    ! [I6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups129246275422532515t_real @ F @ I6 ) )
          = ( groups6591440286371151544t_real
            @ ^ [X4: nat] : ( ln_ln_real @ ( F @ X4 ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_6616_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : X4
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_6617_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S: real,T: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_real @ F @ S )
       => ( ( sums_real @ G @ T )
         => ( ord_less_eq_real @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6618_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_nat @ F @ S )
       => ( ( sums_nat @ G @ T )
         => ( ord_less_eq_nat @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6619_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S: int,T: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_int @ F @ S )
       => ( ( sums_int @ G @ T )
         => ( ord_less_eq_int @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6620_sums__0,axiom,
    ! [F: nat > nat] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_nat )
     => ( sums_nat @ F @ zero_zero_nat ) ) ).

% sums_0
thf(fact_6621_sums__0,axiom,
    ! [F: nat > real] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_real )
     => ( sums_real @ F @ zero_zero_real ) ) ).

% sums_0
thf(fact_6622_sums__0,axiom,
    ! [F: nat > int] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_int )
     => ( sums_int @ F @ zero_zero_int ) ) ).

% sums_0
thf(fact_6623_sums__0,axiom,
    ! [F: nat > complex] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_complex )
     => ( sums_complex @ F @ zero_zero_complex ) ) ).

% sums_0
thf(fact_6624_sums__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( sums_nat
      @ ^ [R4: nat] : ( if_nat @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_nat )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6625_sums__single,axiom,
    ! [I: nat,F: nat > real] :
      ( sums_real
      @ ^ [R4: nat] : ( if_real @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_real )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6626_sums__single,axiom,
    ! [I: nat,F: nat > int] :
      ( sums_int
      @ ^ [R4: nat] : ( if_int @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_int )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6627_sums__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( sums_complex
      @ ^ [R4: nat] : ( if_complex @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_complex )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6628_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B2: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B2 )
       => ( sums_nat
          @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% sums_add
thf(fact_6629_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B2: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B2 )
       => ( sums_int
          @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% sums_add
thf(fact_6630_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B2: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B2 )
       => ( sums_real
          @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% sums_add
thf(fact_6631_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_6632_sums__mult__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
          @ ( times_times_complex @ C @ D ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_6633_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_6634_sums__mult2__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C )
          @ ( times_times_complex @ D @ C ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_6635_sums__mult__D,axiom,
    ! [C: complex,F: nat > complex,A: complex] :
      ( ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
        @ A )
     => ( ( C != zero_zero_complex )
       => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_6636_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
        @ A )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_6637_sums__Suc__imp,axiom,
    ! [F: nat > real,S: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( F @ ( suc @ N ) )
          @ S )
       => ( sums_real @ F @ S ) ) ) ).

% sums_Suc_imp
thf(fact_6638_sums__Suc__imp,axiom,
    ! [F: nat > complex,S: complex] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N: nat] : ( F @ ( suc @ N ) )
          @ S )
       => ( sums_complex @ F @ S ) ) ) ).

% sums_Suc_imp
thf(fact_6639_sums__Suc__iff,axiom,
    ! [F: nat > real,S: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_6640_sums__Suc,axiom,
    ! [F: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L )
     => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6641_sums__Suc,axiom,
    ! [F: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L )
     => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6642_sums__Suc,axiom,
    ! [F: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L )
     => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6643_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > real,S: real] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ( F @ I4 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
          @ S )
        = ( sums_real @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_6644_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > complex,S: complex] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ( F @ I4 )
            = zero_zero_complex ) )
     => ( ( sums_complex
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
          @ S )
        = ( sums_complex @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_6645_sums__finite,axiom,
    ! [N6: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6646_sums__finite,axiom,
    ! [N6: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_complex ) )
       => ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6647_sums__finite,axiom,
    ! [N6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6648_sums__finite,axiom,
    ! [N6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6649_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_int
        @ ^ [R4: nat] : ( if_int @ ( P @ R4 ) @ ( F @ R4 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_6650_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_complex
        @ ^ [R4: nat] : ( if_complex @ ( P @ R4 ) @ ( F @ R4 ) @ zero_zero_complex )
        @ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_6651_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_nat
        @ ^ [R4: nat] : ( if_nat @ ( P @ R4 ) @ ( F @ R4 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_6652_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_real
        @ ^ [R4: nat] : ( if_real @ ( P @ R4 ) @ ( F @ R4 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_6653_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_int
        @ ^ [R4: nat] : ( if_int @ ( member_nat @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_6654_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_complex
        @ ^ [R4: nat] : ( if_complex @ ( member_nat @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_complex )
        @ ( groups2073611262835488442omplex @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_6655_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_nat
        @ ^ [R4: nat] : ( if_nat @ ( member_nat @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_6656_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_real
        @ ^ [R4: nat] : ( if_real @ ( member_nat @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_6657_powser__sums__if,axiom,
    ! [M2: nat,Z3: int] :
      ( sums_int
      @ ^ [N: nat] : ( times_times_int @ ( if_int @ ( N = M2 ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z3 @ N ) )
      @ ( power_power_int @ Z3 @ M2 ) ) ).

% powser_sums_if
thf(fact_6658_powser__sums__if,axiom,
    ! [M2: nat,Z3: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( if_real @ ( N = M2 ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z3 @ N ) )
      @ ( power_power_real @ Z3 @ M2 ) ) ).

% powser_sums_if
thf(fact_6659_powser__sums__if,axiom,
    ! [M2: nat,Z3: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( times_times_complex @ ( if_complex @ ( N = M2 ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z3 @ N ) )
      @ ( power_power_complex @ Z3 @ M2 ) ) ).

% powser_sums_if
thf(fact_6660_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
            @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N ) ) ) ) ) ) ) ).

% ln_series
thf(fact_6661_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( 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_6662_arsinh__def,axiom,
    ( arsinh_real
    = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arsinh_def
thf(fact_6663_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_6664_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M2 ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_6665_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
      = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_6666_central__binomial__lower__bound,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_6667_binomial__Suc__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ ( suc @ N2 ) @ N2 )
      = ( suc @ N2 ) ) ).

% binomial_Suc_n
thf(fact_6668_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_6669_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_6670_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_6671_suminf__zero,axiom,
    ( ( suminf_complex
      @ ^ [N: nat] : zero_zero_complex )
    = zero_zero_complex ) ).

% suminf_zero
thf(fact_6672_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_6673_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_6674_of__real__0,axiom,
    ( ( real_V1803761363581548252l_real @ zero_zero_real )
    = zero_zero_real ) ).

% of_real_0
thf(fact_6675_of__real__0,axiom,
    ( ( real_V4546457046886955230omplex @ zero_zero_real )
    = zero_zero_complex ) ).

% of_real_0
thf(fact_6676_floor__zero,axiom,
    ( ( archim6058952711729229775r_real @ zero_zero_real )
    = zero_zero_int ) ).

% floor_zero
thf(fact_6677_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_6678_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_6679_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_6680_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
      = zero_zero_complex ) ).

% gbinomial_0(2)
thf(fact_6681_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_6682_binomial__1,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
      = N2 ) ).

% binomial_1
thf(fact_6683_binomial__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( binomial @ N2 @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N2 @ K ) ) ).

% binomial_eq_0_iff
thf(fact_6684_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_6685_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_6686_gbinomial__0_I1_J,axiom,
    ! [A: complex] :
      ( ( gbinomial_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_6687_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_6688_of__real__add,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_add
thf(fact_6689_of__real__add,axiom,
    ! [X: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_add
thf(fact_6690_binomial__Suc__Suc,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_6691_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_6692_zero__less__binomial__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
      = ( ord_less_eq_nat @ K @ N2 ) ) ).

% zero_less_binomial_iff
thf(fact_6693_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_6694_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_6695_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_6696_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_6697_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_6698_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_6699_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_6700_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_6701_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_6702_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_6703_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_6704_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_6705_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_6706_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_6707_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_6708_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_6709_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_6710_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_6711_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_6712_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_6713_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_6714_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_6715_Suc__times__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).

% Suc_times_binomial
thf(fact_6716_Suc__times__binomial__eq,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_6717_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_6718_choose__mult__lemma,axiom,
    ! [M2: nat,R2: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K ) @ ( plus_plus_nat @ M2 @ K ) ) @ ( binomial @ ( plus_plus_nat @ M2 @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M2 @ R2 ) @ M2 ) ) ) ).

% choose_mult_lemma
thf(fact_6719_binomial__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R2 @ N2 )
     => ( ord_less_eq_nat @ ( binomial @ N2 @ R2 ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_6720_zero__less__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).

% zero_less_binomial
thf(fact_6721_le__floor__iff,axiom,
    ! [Z3: int,X: real] :
      ( ( ord_less_eq_int @ Z3 @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ) ).

% le_floor_iff
thf(fact_6722_floor__less__iff,axiom,
    ! [X: real,Z3: int] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z3 )
      = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ) ).

% floor_less_iff
thf(fact_6723_Suc__times__binomial__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_6724_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_6725_int__add__floor,axiom,
    ! [Z3: int,X: real] :
      ( ( plus_plus_int @ Z3 @ ( archim6058952711729229775r_real @ X ) )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ) ) ).

% int_add_floor
thf(fact_6726_floor__add__int,axiom,
    ! [X: real,Z3: int] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z3 )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ) ) ).

% floor_add_int
thf(fact_6727_choose__mult,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( times_times_nat @ ( binomial @ N2 @ M2 ) @ ( binomial @ M2 @ K ) )
          = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M2 @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_6728_binomial__Suc__Suc__eq__times,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_6729_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_6730_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_6731_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ K )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_6732_suminf__finite,axiom,
    ! [N6: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( ( suminf_int @ F )
          = ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).

% suminf_finite
thf(fact_6733_suminf__finite,axiom,
    ! [N6: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_complex ) )
       => ( ( suminf_complex @ F )
          = ( groups2073611262835488442omplex @ F @ N6 ) ) ) ) ).

% suminf_finite
thf(fact_6734_suminf__finite,axiom,
    ! [N6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( ( suminf_nat @ F )
          = ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).

% suminf_finite
thf(fact_6735_suminf__finite,axiom,
    ! [N6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( ( suminf_real @ F )
          = ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).

% suminf_finite
thf(fact_6736_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_6737_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_6738_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_6739_binomial__absorption,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
      = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_6740_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_6741_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_6742_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_6743_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
      = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_6744_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_6745_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_6746_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_6747_floor__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I3 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) ) )
           => ( P @ I3 ) ) ) ) ).

% floor_split
thf(fact_6748_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_6749_floor__unique,axiom,
    ! [Z3: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z3 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = Z3 ) ) ) ).

% floor_unique
thf(fact_6750_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_6751_le__mult__floor,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B2 ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ).

% le_mult_floor
thf(fact_6752_less__floor__iff,axiom,
    ! [Z3: int,X: real] :
      ( ( ord_less_int @ Z3 @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z3 ) @ one_one_real ) @ X ) ) ).

% less_floor_iff
thf(fact_6753_floor__le__iff,axiom,
    ! [X: real,Z3: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z3 )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z3 ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_6754_binomial__mono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
       => ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).

% binomial_mono
thf(fact_6755_binomial__maximum_H,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).

% binomial_maximum'
thf(fact_6756_binomial__maximum,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_6757_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K6 @ N2 )
         => ( ord_less_eq_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_6758_binomial__le__pow2,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% binomial_le_pow2
thf(fact_6759_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_6760_choose__reduce__nat,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N2 @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_6761_times__binomial__minus1__eq,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
        = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_6762_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_6763_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_6764_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_6765_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_6766_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M2: nat,A: complex] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( times_times_complex @ ( gbinomial_complex @ A @ M2 ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M2 ) @ K ) )
        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M2 @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_6767_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_6768_floor__divide__lower,axiom,
    ! [Q3: real,P5: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ Q3 ) @ P5 ) ) ).

% floor_divide_lower
thf(fact_6769_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_6770_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K6 @ N2 )
         => ( ord_less_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_6771_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
       => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).

% binomial_strict_mono
thf(fact_6772_central__binomial__odd,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( binomial @ N2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_6773_binomial__addition__formula,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( binomial @ N2 @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_6774_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_6775_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_6776_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_6777_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_6778_floor__divide__upper,axiom,
    ! [Q3: real,P5: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ P5 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_6779_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X4: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_6780_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_6781_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_6782_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_6783_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_6784_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [J2: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J2 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_6785_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J2: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J2 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_6786_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_6787_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_6788_floor__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_6789_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N2 ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_6790_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_6791_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_6792_gbinomial__partial__row__sum,axiom,
    ! [A: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_6793_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_6794_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_6795_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_6796_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] :
                ( if_complex
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_6797_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_6798_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] :
                ( if_real
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_6799_round__altdef,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X4: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X4 ) ) @ ( archim7802044766580827645g_real @ X4 ) @ ( archim6058952711729229775r_real @ X4 ) ) ) ) ).

% round_altdef
thf(fact_6800_gbinomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% gbinomial_r_part_sum
thf(fact_6801_gbinomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% gbinomial_r_part_sum
thf(fact_6802_atMost__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( set_ord_atMost_nat @ X )
        = ( set_ord_atMost_nat @ Y ) )
      = ( X = Y ) ) ).

% atMost_eq_iff
thf(fact_6803_atMost__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( set_ord_atMost_int @ X )
        = ( set_ord_atMost_int @ Y ) )
      = ( X = Y ) ) ).

% atMost_eq_iff
thf(fact_6804_atMost__iff,axiom,
    ! [I: extended_enat,K: extended_enat] :
      ( ( member_Extended_enat @ I @ ( set_or8332593352340944941d_enat @ K ) )
      = ( ord_le2932123472753598470d_enat @ I @ K ) ) ).

% atMost_iff
thf(fact_6805_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
      = ( ord_less_eq_real @ I @ K ) ) ).

% atMost_iff
thf(fact_6806_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
      = ( ord_less_eq_set_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_6807_atMost__iff,axiom,
    ! [I: set_int,K: set_int] :
      ( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K ) )
      = ( ord_less_eq_set_int @ I @ K ) ) ).

% atMost_iff
thf(fact_6808_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
      = ( ord_less_eq_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_6809_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
      = ( ord_less_eq_int @ I @ K ) ) ).

% atMost_iff
thf(fact_6810_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_6811_atMost__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X ) @ ( set_ord_atMost_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6812_atMost__subset__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6813_atMost__subset__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X ) @ ( set_or58775011639299419et_int @ Y ) )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6814_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6815_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6816_frac__of__int,axiom,
    ! [Z3: int] :
      ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z3 ) )
      = zero_zero_real ) ).

% frac_of_int
thf(fact_6817_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H2: set_nat,H3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H2 ) @ ( set_or4236626031148496127et_nat @ H3 ) )
      = ( ~ ( ord_less_eq_set_nat @ L @ H2 )
        | ( ord_less_eq_set_nat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6818_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H2: set_int,H3: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
      = ( ~ ( ord_less_eq_set_int @ L @ H2 )
        | ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6819_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H2: nat,H3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
      = ( ~ ( ord_less_eq_nat @ L @ H2 )
        | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6820_Icc__subset__Iic__iff,axiom,
    ! [L: int,H2: int,H3: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
      = ( ~ ( ord_less_eq_int @ L @ H2 )
        | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6821_Icc__subset__Iic__iff,axiom,
    ! [L: real,H2: real,H3: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
      = ( ~ ( ord_less_eq_real @ L @ H2 )
        | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6822_sum_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6823_sum_OatMost__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6824_sum_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6825_sum_OatMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6826_prod_OatMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6827_prod_OatMost__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6828_prod_OatMost__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6829_prod_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6830_prod_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6831_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_6832_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: extended_enat] :
      ( bot_bo7653980558646680370d_enat
     != ( set_or8332593352340944941d_enat @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6833_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: real] :
      ( bot_bot_set_real
     != ( set_ord_atMost_real @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6834_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: nat] :
      ( bot_bot_set_nat
     != ( set_ord_atMost_nat @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6835_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: int] :
      ( bot_bot_set_int
     != ( set_ord_atMost_int @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6836_infinite__Iic,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_atMost_int @ A ) ) ).

% infinite_Iic
thf(fact_6837_not__Iic__eq__Icc,axiom,
    ! [H3: int,L: int,H2: int] :
      ( ( set_ord_atMost_int @ H3 )
     != ( set_or1266510415728281911st_int @ L @ H2 ) ) ).

% not_Iic_eq_Icc
thf(fact_6838_not__Iic__eq__Icc,axiom,
    ! [H3: real,L: real,H2: real] :
      ( ( set_ord_atMost_real @ H3 )
     != ( set_or1222579329274155063t_real @ L @ H2 ) ) ).

% not_Iic_eq_Icc
thf(fact_6839_atMost__def,axiom,
    ( set_ord_atMost_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6840_atMost__def,axiom,
    ( set_or4236626031148496127et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6841_atMost__def,axiom,
    ( set_or58775011639299419et_int
    = ( ^ [U2: set_int] :
          ( collect_set_int
          @ ^ [X4: set_int] : ( ord_less_eq_set_int @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6842_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6843_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_eq_int @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6844_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_6845_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_6846_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_6847_not__Iic__le__Icc,axiom,
    ! [H2: int,L3: int,H3: int] :
      ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_6848_not__Iic__le__Icc,axiom,
    ! [H2: real,L3: real,H3: real] :
      ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_6849_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K2: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_6850_frac__ge__0,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).

% frac_ge_0
thf(fact_6851_frac__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).

% frac_lt_1
thf(fact_6852_frac__1__eq,axiom,
    ! [X: real] :
      ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( archim2898591450579166408c_real @ X ) ) ).

% frac_1_eq
thf(fact_6853_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_6854_Iic__subset__Iio__iff,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ ( set_or8332593352340944941d_enat @ A ) @ ( set_or8419480210114673929d_enat @ B2 ) )
      = ( ord_le72135733267957522d_enat @ A @ B2 ) ) ).

% Iic_subset_Iio_iff
thf(fact_6855_Iic__subset__Iio__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B2 ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% Iic_subset_Iio_iff
thf(fact_6856_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B2 ) )
      = ( ord_less_nat @ A @ B2 ) ) ).

% Iic_subset_Iio_iff
thf(fact_6857_Iic__subset__Iio__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B2 ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% Iic_subset_Iio_iff
thf(fact_6858_sum__choose__upper,axiom,
    ! [M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( binomial @ K2 @ M2 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ N2 ) @ ( suc @ M2 ) ) ) ).

% sum_choose_upper
thf(fact_6859_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6860_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( G @ zero_zero_nat )
        @ ( groups7108830773950497114d_enat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6861_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6862_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6863_sum__telescope,axiom,
    ! [F: nat > int,I: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_6864_sum__telescope,axiom,
    ! [F: nat > real,I: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_6865_polyfun__eq__coeffs,axiom,
    ! [C: nat > complex,N2: nat,D: nat > complex] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_6866_polyfun__eq__coeffs,axiom,
    ! [C: nat > real,N2: nat,D: nat > real] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_6867_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6868_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( G @ zero_zero_nat )
        @ ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6869_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( G @ zero_zero_nat )
        @ ( groups7961826882256487087d_enat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6870_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6871_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6872_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J2 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_6873_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [J2: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( A @ I3 @ J2 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_6874_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J2: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( A @ I3 @ J2 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_6875_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J2: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J2 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_6876_sum__choose__lower,axiom,
    ! [R2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K2 ) @ K2 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_6877_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J2 ) @ N2 )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ one_one_nat ) @ M2 ) ) ).

% choose_rising_sum(2)
thf(fact_6878_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J2 ) @ N2 )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_6879_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_6880_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > complex,N2: nat,K: nat] :
      ( ! [W: complex] :
          ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ W @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( C @ K )
          = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_6881_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > real,N2: nat,K: nat] :
      ( ! [W: real] :
          ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ W @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( C @ K )
          = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_6882_polyfun__eq__0,axiom,
    ! [C: nat > complex,N2: nat] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = zero_zero_complex ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_0
thf(fact_6883_polyfun__eq__0,axiom,
    ! [C: nat > real,N2: nat] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = zero_zero_real ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_0
thf(fact_6884_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_6885_sum_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_6886_sum_OatMost__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( G @ zero_zero_nat )
        @ ( groups7108830773950497114d_enat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_6887_sum_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_6888_sum_OatMost__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_6889_sum__up__index__split,axiom,
    ! [F: nat > int,M2: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6890_sum__up__index__split,axiom,
    ! [F: nat > extended_enat,M2: nat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups7108830773950497114d_enat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6891_sum__up__index__split,axiom,
    ! [F: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6892_sum__up__index__split,axiom,
    ! [F: nat > real,M2: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6893_prod_OatMost__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_6894_prod_OatMost__shift,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_complex @ ( G @ zero_zero_nat )
        @ ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_6895_prod_OatMost__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( G @ zero_zero_nat )
        @ ( groups7961826882256487087d_enat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_6896_prod_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_6897_prod_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_6898_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_6899_gbinomial__parallel__sum,axiom,
    ! [A: complex,N2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ K2 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) @ one_one_complex ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_6900_gbinomial__parallel__sum,axiom,
    ! [A: real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) @ one_one_real ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_6901_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K2 @ I3 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_6902_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > real,N2: nat] :
      ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K2 @ I3 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_6903_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K2: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K2 @ I3 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_6904_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K2: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K2 @ I3 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_6905_sum__choose__diagonal,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K2 ) @ ( minus_minus_nat @ M2 @ K2 ) )
          @ ( set_ord_atMost_nat @ M2 ) )
        = ( binomial @ ( suc @ N2 ) @ M2 ) ) ) ).

% sum_choose_diagonal
thf(fact_6906_vandermonde,axiom,
    ! [M2: nat,N2: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M2 @ K2 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M2 @ N2 ) @ R2 ) ) ).

% vandermonde
thf(fact_6907_sum__gp__basic,axiom,
    ! [X: int,N2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_6908_sum__gp__basic,axiom,
    ! [X: complex,N2: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_6909_sum__gp__basic,axiom,
    ! [X: real,N2: nat] :
      ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_6910_polyfun__roots__finite,axiom,
    ! [C: nat > complex,K: nat,N2: nat] :
      ( ( ( C @ K )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z6 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = zero_zero_complex ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_6911_polyfun__roots__finite,axiom,
    ! [C: nat > real,K: nat,N2: nat] :
      ( ( ( C @ K )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Z6: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z6 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = zero_zero_real ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_6912_polyfun__finite__roots,axiom,
    ! [C: nat > complex,N2: nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] :
              ( ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) )
              = zero_zero_complex ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
            & ( ( C @ I3 )
             != zero_zero_complex ) ) ) ) ).

% polyfun_finite_roots
thf(fact_6913_polyfun__finite__roots,axiom,
    ! [C: nat > real,N2: nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) )
              = zero_zero_real ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
            & ( ( C @ I3 )
             != zero_zero_real ) ) ) ) ).

% polyfun_finite_roots
thf(fact_6914_polyfun__linear__factor__root,axiom,
    ! [C: nat > int,A: int,N2: nat] :
      ( ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int )
     => ~ ! [B4: nat > int] :
            ~ ! [Z4: int] :
                ( ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_int @ ( minus_minus_int @ Z4 @ A )
                  @ ( groups3539618377306564664at_int
                    @ ^ [I3: nat] : ( times_times_int @ ( B4 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_6915_polyfun__linear__factor__root,axiom,
    ! [C: nat > complex,A: complex,N2: nat] :
      ( ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex )
     => ~ ! [B4: nat > complex] :
            ~ ! [Z4: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
                  @ ( groups2073611262835488442omplex
                    @ ^ [I3: nat] : ( times_times_complex @ ( B4 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_6916_polyfun__linear__factor__root,axiom,
    ! [C: nat > real,A: real,N2: nat] :
      ( ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real )
     => ~ ! [B4: nat > real] :
            ~ ! [Z4: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_real @ ( minus_minus_real @ Z4 @ A )
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( B4 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_6917_polyfun__linear__factor,axiom,
    ! [C: nat > int,N2: nat,A: int] :
    ? [B4: nat > int] :
    ! [Z4: int] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int
        @ ( times_times_int @ ( minus_minus_int @ Z4 @ A )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( B4 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_6918_polyfun__linear__factor,axiom,
    ! [C: nat > complex,N2: nat,A: complex] :
    ? [B4: nat > complex] :
    ! [Z4: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_complex
        @ ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( B4 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_6919_polyfun__linear__factor,axiom,
    ! [C: nat > real,N2: nat,A: real] :
    ? [B4: nat > real] :
    ! [Z4: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_real
        @ ( times_times_real @ ( minus_minus_real @ Z4 @ A )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( B4 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_6920_sum__power__shift,axiom,
    ! [M2: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6921_sum__power__shift,axiom,
    ! [M2: nat,N2: nat,X: complex] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6922_sum__power__shift,axiom,
    ! [M2: nat,N2: nat,X: real] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6923_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K2 @ I3 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.triangle_reindex
thf(fact_6924_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > real,N2: nat] :
      ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K2 @ I3 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.triangle_reindex
thf(fact_6925_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K2: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K2 @ I3 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_6926_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K2: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K2 @ I3 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_6927_binomial,axiom,
    ! [A: nat,B2: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial
thf(fact_6928_sum_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_6929_sum_Oin__pairs__0,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups7108830773950497114d_enat
        @ ^ [I3: nat] : ( plus_p3455044024723400733d_enat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_6930_sum_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_6931_sum_Oin__pairs__0,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_6932_polynomial__product,axiom,
    ! [M2: nat,A: nat > int,N2: nat,B2: nat > int,X: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M2 @ I4 )
         => ( ( A @ I4 )
            = zero_zero_int ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N2 @ J3 )
           => ( ( B2 @ J3 )
              = zero_zero_int ) )
       => ( ( times_times_int
            @ ( groups3539618377306564664at_int
              @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups3539618377306564664at_int
              @ ^ [J2: nat] : ( times_times_int @ ( B2 @ J2 ) @ ( power_power_int @ X @ J2 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3539618377306564664at_int
            @ ^ [R4: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [K2: nat] : ( times_times_int @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R4 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_int @ X @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_6933_polynomial__product,axiom,
    ! [M2: nat,A: nat > complex,N2: nat,B2: nat > complex,X: complex] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M2 @ I4 )
         => ( ( A @ I4 )
            = zero_zero_complex ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N2 @ J3 )
           => ( ( B2 @ J3 )
              = zero_zero_complex ) )
       => ( ( times_times_complex
            @ ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups2073611262835488442omplex
              @ ^ [J2: nat] : ( times_times_complex @ ( B2 @ J2 ) @ ( power_power_complex @ X @ J2 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups2073611262835488442omplex
            @ ^ [R4: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [K2: nat] : ( times_times_complex @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R4 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_complex @ X @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_6934_polynomial__product,axiom,
    ! [M2: nat,A: nat > real,N2: nat,B2: nat > real,X: real] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M2 @ I4 )
         => ( ( A @ I4 )
            = zero_zero_real ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N2 @ J3 )
           => ( ( B2 @ J3 )
              = zero_zero_real ) )
       => ( ( times_times_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups6591440286371151544t_real
              @ ^ [J2: nat] : ( times_times_real @ ( B2 @ J2 ) @ ( power_power_real @ X @ J2 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups6591440286371151544t_real
            @ ^ [R4: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [K2: nat] : ( times_times_real @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R4 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_real @ X @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_6935_prod_Oin__pairs__0,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_6936_prod_Oin__pairs__0,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [I3: nat] : ( times_times_complex @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_6937_prod_Oin__pairs__0,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups7961826882256487087d_enat
        @ ^ [I3: nat] : ( times_7803423173614009249d_enat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_6938_prod_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_6939_prod_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_6940_polyfun__eq__const,axiom,
    ! [C: nat > complex,N2: nat,K: complex] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
           => ( ( C @ X4 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
thf(fact_6941_polyfun__eq__const,axiom,
    ! [C: nat > real,N2: nat,K: real] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
           => ( ( C @ X4 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
thf(fact_6942_binomial__ring,axiom,
    ! [A: complex,B2: complex,N2: nat] :
      ( ( power_power_complex @ ( plus_plus_complex @ A @ B2 ) @ N2 )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K2 ) ) @ ( power_power_complex @ A @ K2 ) ) @ ( power_power_complex @ B2 @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_6943_binomial__ring,axiom,
    ! [A: extended_enat,B2: extended_enat,N2: nat] :
      ( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ A @ B2 ) @ N2 )
      = ( groups7108830773950497114d_enat
        @ ^ [K2: nat] : ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ ( binomial @ N2 @ K2 ) ) @ ( power_8040749407984259932d_enat @ A @ K2 ) ) @ ( power_8040749407984259932d_enat @ B2 @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_6944_binomial__ring,axiom,
    ! [A: int,B2: int,N2: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B2 ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K2 ) ) @ ( power_power_int @ A @ K2 ) ) @ ( power_power_int @ B2 @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_6945_binomial__ring,axiom,
    ! [A: nat,B2: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_6946_binomial__ring,axiom,
    ! [A: real,B2: real,N2: nat] :
      ( ( power_power_real @ ( plus_plus_real @ A @ B2 ) @ N2 )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K2 ) ) @ ( power_power_real @ A @ K2 ) ) @ ( power_power_real @ B2 @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_6947_pochhammer__binomial__sum,axiom,
    ! [A: complex,B2: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ B2 ) @ N2 )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K2 ) ) @ ( comm_s2602460028002588243omplex @ A @ K2 ) ) @ ( comm_s2602460028002588243omplex @ B2 @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_6948_pochhammer__binomial__sum,axiom,
    ! [A: int,B2: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B2 ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K2 ) ) @ ( comm_s4660882817536571857er_int @ A @ K2 ) ) @ ( comm_s4660882817536571857er_int @ B2 @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_6949_pochhammer__binomial__sum,axiom,
    ! [A: real,B2: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B2 ) @ N2 )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K2 ) ) @ ( comm_s7457072308508201937r_real @ A @ K2 ) ) @ ( comm_s7457072308508201937r_real @ B2 @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_6950_polynomial__product__nat,axiom,
    ! [M2: nat,A: nat > nat,N2: nat,B2: nat > nat,X: nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M2 @ I4 )
         => ( ( A @ I4 )
            = zero_zero_nat ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N2 @ J3 )
           => ( ( B2 @ J3 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J2: nat] : ( times_times_nat @ ( B2 @ J2 ) @ ( power_power_nat @ X @ J2 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R4: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R4 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_nat @ X @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_6951_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_6952_sum_Ozero__middle,axiom,
    ! [P5: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P5 )
     => ( ( ord_less_eq_nat @ K @ P5 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J2: nat] : ( if_int @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_int @ ( J2 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P5 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J2: nat] : ( if_int @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6953_sum_Ozero__middle,axiom,
    ! [P5: nat,K: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P5 )
     => ( ( ord_less_eq_nat @ K @ P5 )
       => ( ( groups2073611262835488442omplex
            @ ^ [J2: nat] : ( if_complex @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_complex @ ( J2 = K ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P5 ) )
          = ( groups2073611262835488442omplex
            @ ^ [J2: nat] : ( if_complex @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6954_sum_Ozero__middle,axiom,
    ! [P5: nat,K: nat,G: nat > extended_enat,H2: nat > extended_enat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P5 )
     => ( ( ord_less_eq_nat @ K @ P5 )
       => ( ( groups7108830773950497114d_enat
            @ ^ [J2: nat] : ( if_Extended_enat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_Extended_enat @ ( J2 = K ) @ zero_z5237406670263579293d_enat @ ( H2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P5 ) )
          = ( groups7108830773950497114d_enat
            @ ^ [J2: nat] : ( if_Extended_enat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6955_sum_Ozero__middle,axiom,
    ! [P5: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P5 )
     => ( ( ord_less_eq_nat @ K @ P5 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_nat @ ( J2 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P5 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6956_sum_Ozero__middle,axiom,
    ! [P5: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P5 )
     => ( ( ord_less_eq_nat @ K @ P5 )
       => ( ( groups6591440286371151544t_real
            @ ^ [J2: nat] : ( if_real @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_real @ ( J2 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P5 ) )
          = ( groups6591440286371151544t_real
            @ ^ [J2: nat] : ( if_real @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6957_prod_Ozero__middle,axiom,
    ! [P5: nat,K: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P5 )
     => ( ( ord_less_eq_nat @ K @ P5 )
       => ( ( groups6464643781859351333omplex
            @ ^ [J2: nat] : ( if_complex @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_complex @ ( J2 = K ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P5 ) )
          = ( groups6464643781859351333omplex
            @ ^ [J2: nat] : ( if_complex @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6958_prod_Ozero__middle,axiom,
    ! [P5: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P5 )
     => ( ( ord_less_eq_nat @ K @ P5 )
       => ( ( groups129246275422532515t_real
            @ ^ [J2: nat] : ( if_real @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_real @ ( J2 = K ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P5 ) )
          = ( groups129246275422532515t_real
            @ ^ [J2: nat] : ( if_real @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6959_prod_Ozero__middle,axiom,
    ! [P5: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P5 )
     => ( ( ord_less_eq_nat @ K @ P5 )
       => ( ( groups705719431365010083at_int
            @ ^ [J2: nat] : ( if_int @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_int @ ( J2 = K ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P5 ) )
          = ( groups705719431365010083at_int
            @ ^ [J2: nat] : ( if_int @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6960_prod_Ozero__middle,axiom,
    ! [P5: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P5 )
     => ( ( ord_less_eq_nat @ K @ P5 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_nat @ ( J2 = K ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P5 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P5 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6961_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ A ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_6962_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: real,X: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ A ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K2 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_6963_root__polyfun,axiom,
    ! [N2: nat,Z3: complex,A: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_complex @ Z3 @ N2 )
          = A )
        = ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( if_complex @ ( I3 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I3 = N2 ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z3 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_complex ) ) ) ).

% root_polyfun
thf(fact_6964_root__polyfun,axiom,
    ! [N2: nat,Z3: int,A: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_int @ Z3 @ N2 )
          = A )
        = ( ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( if_int @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I3 = N2 ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z3 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_int ) ) ) ).

% root_polyfun
thf(fact_6965_root__polyfun,axiom,
    ! [N2: nat,Z3: real,A: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_real @ Z3 @ N2 )
          = A )
        = ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( if_real @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I3 = N2 ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z3 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_real ) ) ) ).

% root_polyfun
thf(fact_6966_sum__gp0,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_6967_sum__gp0,axiom,
    ! [X: real,N2: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_6968_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ I3 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
thf(fact_6969_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ I3 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_6970_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ I3 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real ) ) ).

% choose_alternating_linear_sum
thf(fact_6971_gbinomial__sum__nat__pow2,axiom,
    ! [M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M2 @ K2 ) ) @ K2 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M2 ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_6972_gbinomial__sum__nat__pow2,axiom,
    ! [M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ K2 ) ) @ K2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M2 ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_6973_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ A ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A ) @ one_one_complex ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_6974_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: real,X: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ A ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A ) @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_6975_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J2: nat] :
                ( groups3539618377306564664at_int
                @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J2 @ K2 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K2 ) ) @ ( power_power_int @ X @ J2 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J2 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_6976_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J2: nat] :
                ( groups2073611262835488442omplex
                @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J2 @ K2 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K2 ) ) @ ( power_power_complex @ X @ J2 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J2 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_6977_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J2 @ K2 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K2 ) ) @ ( power_power_real @ X @ J2 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J2 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_6978_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_6979_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_sum
thf(fact_6980_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_6981_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real ) ) ).

% choose_alternating_sum
thf(fact_6982_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: nat > complex,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ? [M8: real] :
        ! [Z4: complex] :
          ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z4 ) )
         => ( ord_less_eq_real
            @ ( real_V1022390504157884413omplex
              @ ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) ) )
            @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_6983_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: nat > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ? [M8: real] :
        ! [Z4: real] :
          ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z4 ) )
         => ( ord_less_eq_real
            @ ( real_V7735802525324610683m_real
              @ ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) ) )
            @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_6984_polyfun__diff,axiom,
    ! [N2: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J2: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
                @ ( power_power_int @ X @ J2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_6985_polyfun__diff,axiom,
    ! [N2: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J2: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
                @ ( power_power_complex @ X @ J2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_6986_polyfun__diff,axiom,
    ! [N2: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J2: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
                @ ( power_power_real @ X @ J2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_6987_sin__cos__npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% sin_cos_npi
thf(fact_6988_cos__pi__eq__zero,axiom,
    ! [M2: nat] :
      ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = zero_zero_real ) ).

% cos_pi_eq_zero
thf(fact_6989_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A3: complex,K2: nat] :
          ( if_complex @ ( K2 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K2 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ) ).

% gbinomial_code
thf(fact_6990_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A3: real,K2: nat] :
          ( if_real @ ( K2 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K2 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ) ).

% gbinomial_code
thf(fact_6991_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D5: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_6992_binomial__code,axiom,
    ( binomial
    = ( ^ [N: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K2 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).

% binomial_code
thf(fact_6993_modulo__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M2: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = 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 @ N2 ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = 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 @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ) )
          & ( ( ( 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 @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N2
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N2 @ M2 ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_6994_sgn__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
      = ( sgn_sgn_int @ A ) ) ).

% sgn_sgn
thf(fact_6995_sgn__sgn,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
      = ( sgn_sgn_real @ A ) ) ).

% sgn_sgn
thf(fact_6996_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] : N ) ) ).

% of_nat_id
thf(fact_6997_sin__zero,axiom,
    ( ( sin_real @ zero_zero_real )
    = zero_zero_real ) ).

% sin_zero
thf(fact_6998_sin__zero,axiom,
    ( ( sin_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sin_zero
thf(fact_6999_sgn__0,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_0
thf(fact_7000_sgn__0,axiom,
    ( ( sgn_sgn_int @ zero_zero_int )
    = zero_zero_int ) ).

% sgn_0
thf(fact_7001_sgn__0,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_0
thf(fact_7002_sgn__zero,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_zero
thf(fact_7003_sgn__zero,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_zero
thf(fact_7004_sgn__1,axiom,
    ( ( sgn_sgn_complex @ one_one_complex )
    = one_one_complex ) ).

% sgn_1
thf(fact_7005_sgn__1,axiom,
    ( ( sgn_sgn_int @ one_one_int )
    = one_one_int ) ).

% sgn_1
thf(fact_7006_sgn__1,axiom,
    ( ( sgn_sgn_real @ one_one_real )
    = one_one_real ) ).

% sgn_1
thf(fact_7007_sgn__divide,axiom,
    ! [A: real,B2: real] :
      ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B2 ) ) ) ).

% sgn_divide
thf(fact_7008_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_7009_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_7010_summable__zero,axiom,
    ( summable_nat
    @ ^ [N: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_7011_summable__zero,axiom,
    ( summable_real
    @ ^ [N: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_7012_summable__zero,axiom,
    ( summable_int
    @ ^ [N: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_7013_summable__zero,axiom,
    ( summable_complex
    @ ^ [N: nat] : zero_zero_complex ) ).

% summable_zero
thf(fact_7014_summable__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( summable_nat
      @ ^ [R4: nat] : ( if_nat @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_7015_summable__single,axiom,
    ! [I: nat,F: nat > real] :
      ( summable_real
      @ ^ [R4: nat] : ( if_real @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_7016_summable__single,axiom,
    ! [I: nat,F: nat > int] :
      ( summable_int
      @ ^ [R4: nat] : ( if_int @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_7017_summable__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( summable_complex
      @ ^ [R4: nat] : ( if_complex @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_complex ) ) ).

% summable_single
thf(fact_7018_summable__iff__shift,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_7019_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_7020_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_7021_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_7022_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_7023_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_7024_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_7025_divide__sgn,axiom,
    ! [A: real,B2: real] :
      ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B2 ) )
      = ( times_times_real @ A @ ( sgn_sgn_real @ B2 ) ) ) ).

% divide_sgn
thf(fact_7026_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_7027_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_7028_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_7029_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_7030_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7031_summable__cmult__iff,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7032_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_7033_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_7034_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_nat
        @ ^ [R4: nat] : ( if_nat @ ( member_nat @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite_set
thf(fact_7035_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_real
        @ ^ [R4: nat] : ( if_real @ ( member_nat @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_real ) ) ) ).

% summable_If_finite_set
thf(fact_7036_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_int
        @ ^ [R4: nat] : ( if_int @ ( member_nat @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_int ) ) ) ).

% summable_If_finite_set
thf(fact_7037_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_complex
        @ ^ [R4: nat] : ( if_complex @ ( member_nat @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite_set
thf(fact_7038_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_nat
        @ ^ [R4: nat] : ( if_nat @ ( P @ R4 ) @ ( F @ R4 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite
thf(fact_7039_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_real
        @ ^ [R4: nat] : ( if_real @ ( P @ R4 ) @ ( F @ R4 ) @ zero_zero_real ) ) ) ).

% summable_If_finite
thf(fact_7040_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_int
        @ ^ [R4: nat] : ( if_int @ ( P @ R4 ) @ ( F @ R4 ) @ zero_zero_int ) ) ) ).

% summable_If_finite
thf(fact_7041_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_complex
        @ ^ [R4: nat] : ( if_complex @ ( P @ R4 ) @ ( F @ R4 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite
thf(fact_7042_sgn__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( sgn_sgn_real @ A )
        = one_one_real ) ) ).

% sgn_pos
thf(fact_7043_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_7044_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_7045_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_7046_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_7047_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_7048_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% fact_Suc
thf(fact_7049_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri4449623510593786356d_enat @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ ( suc @ N2 ) ) @ ( semiri4449623510593786356d_enat @ N2 ) ) ) ).

% fact_Suc
thf(fact_7050_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_Suc
thf(fact_7051_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_Suc
thf(fact_7052_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_Suc
thf(fact_7053_sgn__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% sgn_mult_self_eq
thf(fact_7054_sgn__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_mult_self_eq
thf(fact_7055_sin__of__real__pi,axiom,
    ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
    = zero_zero_real ) ).

% sin_of_real_pi
thf(fact_7056_sin__of__real__pi,axiom,
    ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = zero_zero_complex ) ).

% sin_of_real_pi
thf(fact_7057_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_7058_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_7059_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_7060_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_7061_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_7062_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_7063_sin__cos__squared__add3,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_7064_sin__cos__squared__add3,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add3
thf(fact_7065_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_7066_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_7067_sin__cos__squared__add2,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_7068_sin__cos__squared__add2,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_7069_sin__cos__squared__add,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_7070_sin__cos__squared__add,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_7071_cos__of__real__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_of_real_pi_half
thf(fact_7072_cos__of__real__pi__half,axiom,
    ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = zero_zero_complex ) ).

% cos_of_real_pi_half
thf(fact_7073_sin__add,axiom,
    ! [X: real,Y: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% sin_add
thf(fact_7074_sin__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( sin_complex @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( sin_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).

% sin_add
thf(fact_7075_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
     => ( ( sin_real @ X )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_7076_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
        = one_one_complex )
     => ( ( sin_complex @ X )
        = zero_zero_complex ) ) ).

% cos_one_sin_zero
thf(fact_7077_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_7078_fact__mono__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono_nat
thf(fact_7079_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
thf(fact_7080_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
thf(fact_7081_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( sgn_sgn_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
thf(fact_7082_sgn__0__0,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_0_0
thf(fact_7083_sgn__0__0,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_0_0
thf(fact_7084_sgn__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sgn_sgn_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% sgn_zero_iff
thf(fact_7085_sgn__zero__iff,axiom,
    ! [X: complex] :
      ( ( ( sgn_sgn_complex @ X )
        = zero_zero_complex )
      = ( X = zero_zero_complex ) ) ).

% sgn_zero_iff
thf(fact_7086_sgn__mult,axiom,
    ! [A: int,B2: int] :
      ( ( sgn_sgn_int @ ( times_times_int @ A @ B2 ) )
      = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B2 ) ) ) ).

% sgn_mult
thf(fact_7087_sgn__mult,axiom,
    ! [A: real,B2: real] :
      ( ( sgn_sgn_real @ ( times_times_real @ A @ B2 ) )
      = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B2 ) ) ) ).

% sgn_mult
thf(fact_7088_sgn__mult,axiom,
    ! [A: complex,B2: complex] :
      ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B2 ) )
      = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B2 ) ) ) ).

% sgn_mult
thf(fact_7089_same__sgn__sgn__add,axiom,
    ! [B2: int,A: int] :
      ( ( ( sgn_sgn_int @ B2 )
        = ( sgn_sgn_int @ A ) )
     => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B2 ) )
        = ( sgn_sgn_int @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_7090_same__sgn__sgn__add,axiom,
    ! [B2: real,A: real] :
      ( ( ( sgn_sgn_real @ B2 )
        = ( sgn_sgn_real @ A ) )
     => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B2 ) )
        = ( sgn_sgn_real @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_7091_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ N2 )
     != zero_zero_int ) ).

% fact_nonzero
thf(fact_7092_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ N2 )
     != zero_zero_complex ) ).

% fact_nonzero
thf(fact_7093_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri4449623510593786356d_enat @ N2 )
     != zero_z5237406670263579293d_enat ) ).

% fact_nonzero
thf(fact_7094_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ N2 )
     != zero_zero_nat ) ).

% fact_nonzero
thf(fact_7095_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ N2 )
     != zero_zero_real ) ).

% fact_nonzero
thf(fact_7096_summable__const__iff,axiom,
    ! [C: real] :
      ( ( summable_real
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_7097_summable__const__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_complex ) ) ).

% summable_const_iff
thf(fact_7098_summable__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_7099_summable__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_7100_summable__comparison__test_H,axiom,
    ! [G: nat > real,N6: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7101_summable__comparison__test_H,axiom,
    ! [G: nat > real,N6: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7102_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7103_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7104_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7105_cos__diff,axiom,
    ! [X: real,Y: real] :
      ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% cos_diff
thf(fact_7106_cos__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).

% cos_diff
thf(fact_7107_cos__add,axiom,
    ! [X: real,Y: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% cos_add
thf(fact_7108_cos__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).

% cos_add
thf(fact_7109_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_7110_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7111_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_7112_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( ( sin_complex @ X )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_7113_suminf__le,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7114_suminf__le,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7115_suminf__le,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7116_summable__finite,axiom,
    ! [N6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( summable_nat @ F ) ) ) ).

% summable_finite
thf(fact_7117_summable__finite,axiom,
    ! [N6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( summable_real @ F ) ) ) ).

% summable_finite
thf(fact_7118_summable__finite,axiom,
    ! [N6: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( summable_int @ F ) ) ) ).

% summable_finite
thf(fact_7119_summable__finite,axiom,
    ! [N6: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_complex ) )
       => ( summable_complex @ F ) ) ) ).

% summable_finite
thf(fact_7120_fact__less__mono__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono_nat
thf(fact_7121_sgn__not__eq__imp,axiom,
    ! [B2: int,A: int] :
      ( ( ( sgn_sgn_int @ B2 )
       != ( sgn_sgn_int @ A ) )
     => ( ( ( sgn_sgn_int @ A )
         != zero_zero_int )
       => ( ( ( sgn_sgn_int @ B2 )
           != zero_zero_int )
         => ( ( sgn_sgn_int @ A )
            = ( uminus_uminus_int @ ( sgn_sgn_int @ B2 ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_7122_sgn__not__eq__imp,axiom,
    ! [B2: real,A: real] :
      ( ( ( sgn_sgn_real @ B2 )
       != ( sgn_sgn_real @ A ) )
     => ( ( ( sgn_sgn_real @ A )
         != zero_zero_real )
       => ( ( ( sgn_sgn_real @ B2 )
           != zero_zero_real )
         => ( ( sgn_sgn_real @ A )
            = ( uminus_uminus_real @ ( sgn_sgn_real @ B2 ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_7123_sgn__minus__1,axiom,
    ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% sgn_minus_1
thf(fact_7124_sgn__minus__1,axiom,
    ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sgn_minus_1
thf(fact_7125_sgn__minus__1,axiom,
    ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sgn_minus_1
thf(fact_7126_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_zero
thf(fact_7127_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_zero
thf(fact_7128_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_zero
thf(fact_7129_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_gt_zero
thf(fact_7130_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_gt_zero
thf(fact_7131_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_gt_zero
thf(fact_7132_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_7133_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_7134_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_7135_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N3: nat,L4: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_sgnE
thf(fact_7136_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_1
thf(fact_7137_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_1
thf(fact_7138_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_1
thf(fact_7139_fact__mono,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_mono
thf(fact_7140_fact__mono,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono
thf(fact_7141_fact__mono,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_mono
thf(fact_7142_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_7143_summable__mult__D,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
     => ( ( C != zero_zero_complex )
       => ( summable_complex @ F ) ) ) ).

% summable_mult_D
thf(fact_7144_fact__dvd,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M2 ) ) ) ).

% fact_dvd
thf(fact_7145_fact__dvd,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M2 ) ) ) ).

% fact_dvd
thf(fact_7146_fact__dvd,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ M2 ) ) ) ).

% fact_dvd
thf(fact_7147_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7148_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7149_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7150_suminf__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
          = ( suminf_nat
            @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7151_suminf__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
          = ( suminf_int
            @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7152_suminf__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7153_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7154_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7155_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7156_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7157_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7158_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7159_suminf__pos,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_pos
thf(fact_7160_suminf__pos,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_pos
thf(fact_7161_suminf__pos,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_pos
thf(fact_7162_fact__ge__Suc__0__nat,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_Suc_0_nat
thf(fact_7163_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_7164_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_7165_dvd__fact,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( dvd_dvd_nat @ M2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% dvd_fact
thf(fact_7166_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).

% summable_0_powser
thf(fact_7167_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).

% summable_0_powser
thf(fact_7168_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N: nat] : ( times_times_int @ ( F @ N ) @ ( power_power_int @ zero_zero_int @ N ) ) ) ).

% summable_zero_power'
thf(fact_7169_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).

% summable_zero_power'
thf(fact_7170_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).

% summable_zero_power'
thf(fact_7171_summable__powser__split__head,axiom,
    ! [F: nat > real,Z3: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z3 @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z3 @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_7172_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z3: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z3 @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z3 @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_7173_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z3: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z3 @ N ) ) )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z3 @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_7174_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z3: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z3 @ N ) ) )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z3 @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_7175_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M2: nat,Z3: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N @ M2 ) ) @ ( power_power_real @ Z3 @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z3 @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7176_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M2: nat,Z3: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N @ M2 ) ) @ ( power_power_complex @ Z3 @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z3 @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7177_fact__less__mono,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N2 )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_7178_fact__less__mono,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_7179_fact__less__mono,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N2 )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_7180_sin__times__sin,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z3 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z3 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z3 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_7181_sin__times__sin,axiom,
    ! [W2: real,Z3: real] :
      ( ( times_times_real @ ( sin_real @ W2 ) @ ( sin_real @ Z3 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z3 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_7182_sin__times__cos,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( cos_complex @ Z3 ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z3 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z3 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_7183_sin__times__cos,axiom,
    ! [W2: real,Z3: real] :
      ( ( times_times_real @ ( sin_real @ W2 ) @ ( cos_real @ Z3 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z3 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_7184_cos__times__sin,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( sin_complex @ Z3 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z3 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z3 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_7185_cos__times__sin,axiom,
    ! [W2: real,Z3: real] :
      ( ( times_times_real @ ( cos_real @ W2 ) @ ( sin_real @ Z3 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z3 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_7186_sin__plus__sin,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z3 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_7187_sin__plus__sin,axiom,
    ! [W2: real,Z3: real] :
      ( ( plus_plus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z3 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_7188_sin__diff__sin,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z3 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_7189_sin__diff__sin,axiom,
    ! [W2: real,Z3: real] :
      ( ( minus_minus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z3 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_7190_cos__diff__cos,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z3 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z3 @ W2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_7191_cos__diff__cos,axiom,
    ! [W2: real,Z3: real] :
      ( ( minus_minus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z3 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z3 @ W2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_7192_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7193_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7194_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7195_fact__mod,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M2 ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_7196_fact__mod,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M2 ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_7197_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_7198_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_7199_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_7200_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_7201_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7202_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7203_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7204_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7205_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7206_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7207_minus__sin__cos__eq,axiom,
    ! [X: real] :
      ( ( uminus_uminus_real @ ( sin_real @ X ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_7208_minus__sin__cos__eq,axiom,
    ! [X: complex] :
      ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
      = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_7209_suminf__le__const,axiom,
    ! [F: nat > int,X: int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7210_suminf__le__const,axiom,
    ! [F: nat > nat,X: nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7211_suminf__le__const,axiom,
    ! [F: nat > real,X: real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7212_fact__diff__Suc,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_7213_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_7214_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_7215_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X4: int] : ( if_int @ ( X4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_7216_sgn__if,axiom,
    ( sgn_sgn_real
    = ( ^ [X4: real] : ( if_real @ ( X4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_if
thf(fact_7217_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_7218_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R2 @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R2 ) ) ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_7219_binomial__fact__lemma,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
        = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% binomial_fact_lemma
thf(fact_7220_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7221_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7222_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7223_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_7224_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_7225_bounded__imp__summable,axiom,
    ! [A: nat > int,B: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B )
       => ( summable_int @ A ) ) ) ).

% bounded_imp_summable
thf(fact_7226_bounded__imp__summable,axiom,
    ! [A: nat > nat,B: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B )
       => ( summable_nat @ A ) ) ) ).

% bounded_imp_summable
thf(fact_7227_bounded__imp__summable,axiom,
    ! [A: nat > real,B: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B )
       => ( summable_real @ A ) ) ) ).

% bounded_imp_summable
thf(fact_7228_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7229_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% choose_dvd
thf(fact_7230_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% choose_dvd
thf(fact_7231_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% choose_dvd
thf(fact_7232_fact__eq__fact__times,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( semiri1408675320244567234ct_nat @ M2 )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M2 ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_7233_sin__expansion__lemma,axiom,
    ! [X: real,M2: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_7234_sum__le__suminf,axiom,
    ! [F: nat > int,I6: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I6 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7235_sum__le__suminf,axiom,
    ! [F: nat > nat,I6: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I6 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7236_sum__le__suminf,axiom,
    ! [F: nat > real,I6: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I6 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7237_cos__expansion__lemma,axiom,
    ! [X: real,M2: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_7238_binomial__altdef__nat,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_7239_suminf__split__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real @ F )
        = ( plus_plus_real
          @ ( suminf_real
            @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
          @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_7240_suminf__minus__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_7241_fact__div__fact,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M2 ) ) ) ) ).

% fact_div_fact
thf(fact_7242_sum__less__suminf,axiom,
    ! [F: nat > int,N2: nat] :
      ( ( summable_int @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_int @ zero_zero_int @ ( F @ M3 ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7243_sum__less__suminf,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( summable_nat @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M3 ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7244_sum__less__suminf,axiom,
    ! [F: nat > real,N2: nat] :
      ( ( summable_real @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_real @ zero_zero_real @ ( F @ M3 ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7245_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z3: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z3 @ N ) ) )
     => ( ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z3 @ N ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z3 @ N ) ) )
            @ Z3 ) ) ) ) ).

% powser_split_head(1)
thf(fact_7246_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z3: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z3 @ N ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z3 @ N ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z3 @ N ) ) )
            @ Z3 ) ) ) ) ).

% powser_split_head(1)
thf(fact_7247_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z3: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z3 @ N ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z3 @ N ) ) )
          @ Z3 )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z3 @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7248_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z3: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z3 @ N ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z3 @ N ) ) )
          @ Z3 )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z3 @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7249_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E2: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N9: nat] :
              ~ ! [M5: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M5 )
                 => ! [N7: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M5 @ N7 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_7250_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E2: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N9: nat] :
              ~ ! [M5: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M5 )
                 => ! [N7: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M5 @ N7 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_7251_suminf__exist__split,axiom,
    ! [R2: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_real @ F )
       => ? [N9: nat] :
          ! [N7: nat] :
            ( ( ord_less_eq_nat @ N9 @ N7 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N7 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_7252_suminf__exist__split,axiom,
    ! [R2: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_complex @ F )
       => ? [N9: nat] :
          ! [N7: nat] :
            ( ( ord_less_eq_nat @ N9 @ N7 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N7 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_7253_sin__pi__divide__n__ge__0,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_7254_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_7255_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M: nat] : ( if_complex @ ( M = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7256_fact__num__eq__if,axiom,
    ( semiri4449623510593786356d_enat
    = ( ^ [M: nat] : ( if_Extended_enat @ ( M = zero_zero_nat ) @ one_on7984719198319812577d_enat @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4449623510593786356d_enat @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7257_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M: nat] : ( if_int @ ( M = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7258_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M: nat] : ( if_nat @ ( M = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7259_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M: nat] : ( if_real @ ( M = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7260_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri5044797733671781792omplex @ N2 )
        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7261_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri4449623510593786356d_enat @ N2 )
        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( semiri4449623510593786356d_enat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7262_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1406184849735516958ct_int @ N2 )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7263_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1408675320244567234ct_nat @ N2 )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7264_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri2265585572941072030t_real @ N2 )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7265_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_7266_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_7267_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_7268_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_7269_cos__times__cos,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z3 ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z3 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z3 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_7270_cos__times__cos,axiom,
    ! [W2: real,Z3: real] :
      ( ( times_times_real @ ( cos_real @ W2 ) @ ( cos_real @ Z3 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z3 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_7271_cos__plus__cos,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z3 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_7272_cos__plus__cos,axiom,
    ! [W2: real,Z3: real] :
      ( ( plus_plus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z3 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_7273_summable__ratio__test,axiom,
    ! [C: real,N6: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_7274_summable__ratio__test,axiom,
    ! [C: real,N6: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_7275_sum__less__suminf2,axiom,
    ! [F: nat > int,N2: nat,I: nat] :
      ( ( summable_int @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M3 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7276_sum__less__suminf2,axiom,
    ! [F: nat > nat,N2: nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M3 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
           => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7277_sum__less__suminf2,axiom,
    ! [F: nat > real,N2: nat,I: nat] :
      ( ( summable_real @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M3 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
           => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7278_gbinomial__pochhammer_H,axiom,
    ( gbinomial_complex
    = ( ^ [A3: complex,K2: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_7279_gbinomial__pochhammer_H,axiom,
    ( gbinomial_real
    = ( ^ [A3: real,K2: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_7280_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > nat > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ X @ M ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_7281_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > real > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ X @ M ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_7282_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > int > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ X @ M ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_7283_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > complex > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ X @ M ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).

% Maclaurin_zero
thf(fact_7284_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > extended_enat > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_z5237406670263579293d_enat ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ X @ M ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_z5237406670263579293d_enat ) ) ) ) ).

% Maclaurin_zero
thf(fact_7285_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M: nat] : ( times_times_real @ ( cos_coeff @ M ) @ ( power_power_real @ X @ M ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_7286_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( 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
                  @ ^ [M: nat] : ( times_times_real @ ( cos_coeff @ M ) @ ( power_power_real @ X @ M ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_7287_sin__pi__divide__n__gt__0,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_7288_gbinomial__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( divide_divide_real
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7289_gbinomial__Suc,axiom,
    ! [A: int,K: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7290_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7291_divide__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M2: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = 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 @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = 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 @ N2 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N2 ) ) ) )
          & ( ( ( 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 @ N2 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M2 @ N2 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N2 @ M2 ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_7292_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( 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
                  @ ^ [M: nat] : ( times_times_real @ ( sin_coeff @ M ) @ ( power_power_real @ X @ M ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_7293_listrel1p__def,axiom,
    ( listrel1p_nat
    = ( ^ [R4: nat > nat > $o,Xs3: list_nat,Ys3: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel1_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R4 ) ) ) ) ) ) ).

% listrel1p_def
thf(fact_7294_listrel1p__def,axiom,
    ( listrel1p_int
    = ( ^ [R4: int > int > $o,Xs3: list_int,Ys3: list_int] : ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys3 ) @ ( listrel1_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R4 ) ) ) ) ) ) ).

% listrel1p_def
thf(fact_7295_sin__x__sin__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P6: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) ) @ ( power_power_complex @ X @ N ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ).

% sin_x_sin_y
thf(fact_7296_sin__x__sin__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P6: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) ) @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ).

% sin_x_sin_y
thf(fact_7297_sums__cos__x__plus__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P6: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_complex @ X @ N ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N ) ) ) @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% sums_cos_x_plus_y
thf(fact_7298_sums__cos__x__plus__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P6: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N ) ) ) @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% sums_cos_x_plus_y
thf(fact_7299_cos__x__cos__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P6: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_complex @ X @ N ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ).

% cos_x_cos_y
thf(fact_7300_cos__x__cos__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P6: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ).

% cos_x_cos_y
thf(fact_7301_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ).

% sin_coeff_def
thf(fact_7302_scaleR__cancel__right,axiom,
    ! [A: real,X: real,B2: real] :
      ( ( ( real_V1485227260804924795R_real @ A @ X )
        = ( real_V1485227260804924795R_real @ B2 @ X ) )
      = ( ( A = B2 )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_cancel_right
thf(fact_7303_scaleR__cancel__right,axiom,
    ! [A: real,X: complex,B2: real] :
      ( ( ( real_V2046097035970521341omplex @ A @ X )
        = ( real_V2046097035970521341omplex @ B2 @ X ) )
      = ( ( A = B2 )
        | ( X = zero_zero_complex ) ) ) ).

% scaleR_cancel_right
thf(fact_7304_scaleR__zero__right,axiom,
    ! [A: real] :
      ( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% scaleR_zero_right
thf(fact_7305_scaleR__zero__right,axiom,
    ! [A: real] :
      ( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% scaleR_zero_right
thf(fact_7306_scaleR__eq__0__iff,axiom,
    ! [A: real,X: real] :
      ( ( ( real_V1485227260804924795R_real @ A @ X )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_eq_0_iff
thf(fact_7307_scaleR__eq__0__iff,axiom,
    ! [A: real,X: complex] :
      ( ( ( real_V2046097035970521341omplex @ A @ X )
        = zero_zero_complex )
      = ( ( A = zero_zero_real )
        | ( X = zero_zero_complex ) ) ) ).

% scaleR_eq_0_iff
thf(fact_7308_scaleR__zero__left,axiom,
    ! [X: real] :
      ( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
      = zero_zero_real ) ).

% scaleR_zero_left
thf(fact_7309_scaleR__zero__left,axiom,
    ! [X: complex] :
      ( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
      = zero_zero_complex ) ).

% scaleR_zero_left
thf(fact_7310_scaleR__eq__iff,axiom,
    ! [B2: real,U: real,A: real] :
      ( ( ( plus_plus_real @ B2 @ ( real_V1485227260804924795R_real @ U @ A ) )
        = ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U @ B2 ) ) )
      = ( ( A = B2 )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
thf(fact_7311_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_7312_scaleR__collapse,axiom,
    ! [U: real,A: real] :
      ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
      = A ) ).

% scaleR_collapse
thf(fact_7313_scaleR__half__double,axiom,
    ! [A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
      = A ) ).

% scaleR_half_double
thf(fact_7314_scaleR__right__imp__eq,axiom,
    ! [X: real,A: real,B2: real] :
      ( ( X != zero_zero_real )
     => ( ( ( real_V1485227260804924795R_real @ A @ X )
          = ( real_V1485227260804924795R_real @ B2 @ X ) )
       => ( A = B2 ) ) ) ).

% scaleR_right_imp_eq
thf(fact_7315_scaleR__right__imp__eq,axiom,
    ! [X: complex,A: real,B2: real] :
      ( ( X != zero_zero_complex )
     => ( ( ( real_V2046097035970521341omplex @ A @ X )
          = ( real_V2046097035970521341omplex @ B2 @ X ) )
       => ( A = B2 ) ) ) ).

% scaleR_right_imp_eq
thf(fact_7316_scaleR__right__distrib,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).

% scaleR_right_distrib
thf(fact_7317_scaleR__left_Oadd,axiom,
    ! [X: real,Y: real,Xa2: real] :
      ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y ) @ Xa2 )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa2 ) @ ( real_V1485227260804924795R_real @ Y @ Xa2 ) ) ) ).

% scaleR_left.add
thf(fact_7318_scaleR__left__distrib,axiom,
    ! [A: real,B2: real,X: real] :
      ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B2 ) @ X )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B2 @ X ) ) ) ).

% scaleR_left_distrib
thf(fact_7319_scaleR__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B2 @ C ) ) ) ) ).

% scaleR_right_mono_neg
thf(fact_7320_scaleR__right__mono,axiom,
    ! [A: real,B2: real,X: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B2 @ X ) ) ) ) ).

% scaleR_right_mono
thf(fact_7321_scaleR__le__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% scaleR_le_cancel_left
thf(fact_7322_scaleR__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B2 ) )
        = ( ord_less_eq_real @ B2 @ A ) ) ) ).

% scaleR_le_cancel_left_neg
thf(fact_7323_scaleR__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B2 ) )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% scaleR_le_cancel_left_pos
thf(fact_7324_scaleR__left__mono,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ) ).

% scaleR_left_mono
thf(fact_7325_scaleR__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B2 ) ) ) ) ).

% scaleR_left_mono_neg
thf(fact_7326_eq__vector__fraction__iff,axiom,
    ! [X: real,U: real,V: real,A: real] :
      ( ( X
        = ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
      = ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V1485227260804924795R_real @ V @ X )
            = ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).

% eq_vector_fraction_iff
thf(fact_7327_eq__vector__fraction__iff,axiom,
    ! [X: complex,U: real,V: real,A: complex] :
      ( ( X
        = ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A ) )
      = ( ( ( V = zero_zero_real )
         => ( X = zero_zero_complex ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V2046097035970521341omplex @ V @ X )
            = ( real_V2046097035970521341omplex @ U @ A ) ) ) ) ) ).

% eq_vector_fraction_iff
thf(fact_7328_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: real,X: real] :
      ( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
        = X )
      = ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V1485227260804924795R_real @ U @ A )
            = ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).

% vector_fraction_eq_iff
thf(fact_7329_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: complex,X: complex] :
      ( ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A )
        = X )
      = ( ( ( V = zero_zero_real )
         => ( X = zero_zero_complex ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V2046097035970521341omplex @ U @ A )
            = ( real_V2046097035970521341omplex @ V @ X ) ) ) ) ) ).

% vector_fraction_eq_iff
thf(fact_7330_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B2 @ A ) @ E2 ) @ D ) ) ) ).

% Real_Vector_Spaces.le_add_iff2
thf(fact_7331_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B2: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% Real_Vector_Spaces.le_add_iff1
thf(fact_7332_zero__le__scaleR__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
thf(fact_7333_scaleR__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
thf(fact_7334_scaleR__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B2 ) ) ) ) ).

% scaleR_nonpos_nonpos
thf(fact_7335_scaleR__nonpos__nonneg,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).

% scaleR_nonpos_nonneg
thf(fact_7336_scaleR__nonneg__nonpos,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).

% scaleR_nonneg_nonpos
thf(fact_7337_scaleR__nonneg__nonneg,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 @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).

% scaleR_nonneg_nonneg
thf(fact_7338_split__scaleR__pos__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B2 ) ) ) ).

% split_scaleR_pos_le
thf(fact_7339_split__scaleR__neg__le,axiom,
    ! [A: real,X: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ X @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
     => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).

% split_scaleR_neg_le
thf(fact_7340_scaleR__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B2 @ D ) ) ) ) ) ) ).

% scaleR_mono'
thf(fact_7341_scaleR__mono,axiom,
    ! [A: real,B2: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ X )
           => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B2 @ Y ) ) ) ) ) ) ).

% scaleR_mono
thf(fact_7342_scaleR__left__le__one__le,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).

% scaleR_left_le_one_le
thf(fact_7343_scaleR__2,axiom,
    ! [X: real] :
      ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
      = ( plus_plus_real @ X @ X ) ) ).

% scaleR_2
thf(fact_7344_sin__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( sin_coeff @ ( suc @ N2 ) )
      = ( divide_divide_real @ ( cos_coeff @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).

% sin_coeff_Suc
thf(fact_7345_cos__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( cos_coeff @ ( suc @ N2 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).

% cos_coeff_Suc
thf(fact_7346_summable__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( summable_real
        @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_7347_diffs__equiv,axiom,
    ! [C: nat > complex,X: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( C @ N ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).

% diffs_equiv
thf(fact_7348_diffs__equiv,axiom,
    ! [C: nat > real,X: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( C @ N ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% diffs_equiv
thf(fact_7349_tan__double,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_7350_tan__double,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_7351_tan__half,axiom,
    ( tan_complex
    = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_7352_tan__half,axiom,
    ( tan_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_7353_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D4: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z6: int] :
                ( ( ord_less_eq_int @ D4 @ Z6 )
                & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_7354_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_7355_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_7356_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_7357_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_7358_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_7359_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_7360_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_7361_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_7362_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_7363_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_7364_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_7365_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_7366_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_7367_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_7368_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_7369_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_7370_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_7371_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_7372_abs__add__abs,axiom,
    ! [A: int,B2: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_add_abs
thf(fact_7373_abs__add__abs,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_add_abs
thf(fact_7374_abs__divide,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_divide
thf(fact_7375_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_7376_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_7377_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_7378_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_7379_abs__dvd__iff,axiom,
    ! [M2: real,K: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M2 ) @ K )
      = ( dvd_dvd_real @ M2 @ K ) ) ).

% abs_dvd_iff
thf(fact_7380_abs__dvd__iff,axiom,
    ! [M2: int,K: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M2 ) @ K )
      = ( dvd_dvd_int @ M2 @ K ) ) ).

% abs_dvd_iff
thf(fact_7381_dvd__abs__iff,axiom,
    ! [M2: real,K: real] :
      ( ( dvd_dvd_real @ M2 @ ( abs_abs_real @ K ) )
      = ( dvd_dvd_real @ M2 @ K ) ) ).

% dvd_abs_iff
thf(fact_7382_dvd__abs__iff,axiom,
    ! [M2: int,K: int] :
      ( ( dvd_dvd_int @ M2 @ ( abs_abs_int @ K ) )
      = ( dvd_dvd_int @ M2 @ K ) ) ).

% dvd_abs_iff
thf(fact_7383_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% abs_of_nat
thf(fact_7384_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% abs_of_nat
thf(fact_7385_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
      = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).

% of_int_abs
thf(fact_7386_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
      = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).

% of_int_abs
thf(fact_7387_tan__zero,axiom,
    ( ( tan_real @ zero_zero_real )
    = zero_zero_real ) ).

% tan_zero
thf(fact_7388_tan__zero,axiom,
    ( ( tan_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% tan_zero
thf(fact_7389_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% abs_bool_eq
thf(fact_7390_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_7391_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_7392_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_7393_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_7394_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_7395_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_7396_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_7397_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_7398_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_7399_sum__abs,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_7400_sum__abs,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_7401_divide__le__0__abs__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A @ zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_7402_zero__le__divide__abs__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        | ( B2 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_7403_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_7404_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_7405_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_7406_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_7407_sgn__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% sgn_abs
thf(fact_7408_sgn__abs,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
      = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).

% sgn_abs
thf(fact_7409_sgn__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_abs
thf(fact_7410_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_7411_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
      = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_7412_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_7413_sum__abs__ge__zero,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_7414_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_7415_zero__less__power__abs__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
      = ( ( A != zero_zero_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7416_zero__less__power__abs__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) )
      = ( ( A != zero_zero_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7417_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_7418_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_7419_norm__of__real__addn,axiom,
    ! [X: real,B2: num] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B2 ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B2 ) ) ) ) ).

% norm_of_real_addn
thf(fact_7420_norm__of__real__addn,axiom,
    ! [X: real,B2: num] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B2 ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B2 ) ) ) ) ).

% norm_of_real_addn
thf(fact_7421_abs__minus__commute,axiom,
    ! [A: int,B2: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) )
      = ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).

% abs_minus_commute
thf(fact_7422_abs__minus__commute,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) )
      = ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).

% abs_minus_commute
thf(fact_7423_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_7424_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_7425_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( ( abs_abs_real @ L )
        = ( abs_abs_real @ K ) )
     => ( dvd_dvd_real @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_7426_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( ( abs_abs_int @ L )
        = ( abs_abs_int @ K ) )
     => ( dvd_dvd_int @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_7427_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_7428_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_7429_abs__mult,axiom,
    ! [A: int,B2: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_mult
thf(fact_7430_abs__mult,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_mult
thf(fact_7431_abs__mult,axiom,
    ! [A: complex,B2: complex] :
      ( ( abs_abs_complex @ ( times_times_complex @ A @ B2 ) )
      = ( times_times_complex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B2 ) ) ) ).

% abs_mult
thf(fact_7432_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_7433_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_7434_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_7435_abs__le__D1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% abs_le_D1
thf(fact_7436_abs__le__D1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% abs_le_D1
thf(fact_7437_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_7438_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_7439_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_7440_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_7441_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7442_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7443_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_7444_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_7445_abs__triangle__ineq,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_7446_abs__triangle__ineq,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_7447_abs__mult__less,axiom,
    ! [A: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B2 ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7448_abs__mult__less,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7449_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7450_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7451_abs__triangle__ineq3,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_7452_abs__triangle__ineq3,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_7453_abs__triangle__ineq2,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_7454_abs__triangle__ineq2,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_7455_nonzero__abs__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_7456_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_7457_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_7458_abs__le__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
      = ( ( ord_less_eq_real @ A @ B2 )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_7459_abs__le__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
      = ( ( ord_less_eq_int @ A @ B2 )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_7460_abs__le__D2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).

% abs_le_D2
thf(fact_7461_abs__le__D2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% abs_le_D2
thf(fact_7462_abs__leI,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
       => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 ) ) ) ).

% abs_leI
thf(fact_7463_abs__leI,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 ) ) ) ).

% abs_leI
thf(fact_7464_abs__less__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B2 )
      = ( ( ord_less_int @ A @ B2 )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_7465_abs__less__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ B2 )
      = ( ( ord_less_real @ A @ B2 )
        & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_7466_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_int
    = ( ^ [K2: int] : ( times_times_int @ K2 @ ( sgn_sgn_int @ K2 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_7467_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_real
    = ( ^ [K2: real] : ( times_times_real @ K2 @ ( sgn_sgn_real @ K2 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_7468_abs__mult__sgn,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_7469_abs__mult__sgn,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_7470_abs__mult__sgn,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_7471_sgn__mult__abs,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_7472_sgn__mult__abs,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_7473_sgn__mult__abs,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_7474_mult__sgn__abs,axiom,
    ! [X: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_7475_mult__sgn__abs,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ X ) @ ( abs_abs_real @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_7476_same__sgn__abs__add,axiom,
    ! [B2: int,A: int] :
      ( ( ( sgn_sgn_int @ B2 )
        = ( sgn_sgn_int @ A ) )
     => ( ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) )
        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).

% same_sgn_abs_add
thf(fact_7477_same__sgn__abs__add,axiom,
    ! [B2: real,A: real] :
      ( ( ( sgn_sgn_real @ B2 )
        = ( sgn_sgn_real @ A ) )
     => ( ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) )
        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).

% same_sgn_abs_add
thf(fact_7478_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_7479_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_7480_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_7481_abs__eq__mult,axiom,
    ! [A: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          | ( ord_less_eq_real @ A @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B2 )
          | ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_7482_abs__eq__mult,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B2 )
          | ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_7483_abs__eq__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( ( abs_abs_real @ A )
        = B2 )
      = ( ( ord_less_eq_real @ zero_zero_real @ B2 )
        & ( ( A = B2 )
          | ( A
            = ( uminus_uminus_real @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7484_abs__eq__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( ( abs_abs_int @ A )
        = B2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ B2 )
        & ( ( A = B2 )
          | ( A
            = ( uminus_uminus_int @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7485_eq__abs__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( abs_abs_real @ B2 ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ( B2 = A )
          | ( B2
            = ( uminus_uminus_real @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7486_eq__abs__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( abs_abs_int @ B2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B2 = A )
          | ( B2
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7487_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_7488_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_7489_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_7490_zero__le__power__abs,axiom,
    ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_7491_zero__le__power__abs,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_7492_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_7493_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_7494_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_7495_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_7496_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_7497_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_7498_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_7499_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_7500_abs__triangle__ineq4,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_7501_abs__triangle__ineq4,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_7502_abs__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7503_abs__diff__triangle__ineq,axiom,
    ! [A: int,B2: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7504_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_7505_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_7506_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_7507_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_7508_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_7509_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_7510_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_7511_of__int__leD,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% of_int_leD
thf(fact_7512_of__int__leD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_7513_of__int__lessD,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% of_int_lessD
thf(fact_7514_of__int__lessD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_7515_sgn__power__injE,axiom,
    ! [A: real,N2: nat,X: real,B2: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B2 ) @ ( power_power_real @ ( abs_abs_real @ B2 ) @ N2 ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( A = B2 ) ) ) ) ).

% sgn_power_injE
thf(fact_7516_round__diff__minimal,axiom,
    ! [Z3: real,M2: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z3 ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ ( ring_1_of_int_real @ M2 ) ) ) ) ).

% round_diff_minimal
thf(fact_7517_diffs__def,axiom,
    ( diffs_complex
    = ( ^ [C3: nat > complex,N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_7518_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C3: nat > real,N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_7519_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C3: nat > int,N: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_7520_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_7521_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_7522_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_7523_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_7524_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
         => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7525_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X: int] :
      ( ! [X5: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7526_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_7527_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_7528_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_7529_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_7530_power__mono__even,axiom,
    ! [N2: nat,A: real,B2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B2 @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7531_power__mono__even,axiom,
    ! [N2: nat,A: int,B2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B2 @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7532_convex__sum__bound__le,axiom,
    ! [I6: set_Extended_enat,X: extended_enat > real,A: extended_enat > real,B2: real,Delta: real] :
      ( ! [I4: extended_enat] :
          ( ( member_Extended_enat @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
     => ( ( ( groups4148127829035722712t_real @ X @ I6 )
          = one_one_real )
       => ( ! [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups4148127829035722712t_real
                  @ ^ [I3: extended_enat] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7533_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > real,A: real > real,B2: real,Delta: real] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X @ I6 )
          = one_one_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7534_convex__sum__bound__le,axiom,
    ! [I6: set_set_nat,X: set_nat > real,A: set_nat > real,B2: real,Delta: real] :
      ( ! [I4: set_nat] :
          ( ( member_set_nat @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
     => ( ( ( groups5107569545109728110t_real @ X @ I6 )
          = one_one_real )
       => ( ! [I4: set_nat] :
              ( ( member_set_nat @ I4 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups5107569545109728110t_real
                  @ ^ [I3: set_nat] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7535_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > real,A: int > real,B2: real,Delta: real] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X @ I6 )
          = one_one_real )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7536_convex__sum__bound__le,axiom,
    ! [I6: set_Extended_enat,X: extended_enat > int,A: extended_enat > int,B2: int,Delta: int] :
      ( ! [I4: extended_enat] :
          ( ( member_Extended_enat @ I4 @ I6 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I4 ) ) )
     => ( ( ( groups2025484359314973016at_int @ X @ I6 )
          = one_one_int )
       => ( ! [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I6 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I4 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups2025484359314973016at_int
                  @ ^ [I3: extended_enat] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7537_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > int,A: real > int,B2: int,Delta: int] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I6 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I4 ) ) )
     => ( ( ( groups1932886352136224148al_int @ X @ I6 )
          = one_one_int )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I4 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups1932886352136224148al_int
                  @ ^ [I3: real] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7538_convex__sum__bound__le,axiom,
    ! [I6: set_set_nat,X: set_nat > int,A: set_nat > int,B2: int,Delta: int] :
      ( ! [I4: set_nat] :
          ( ( member_set_nat @ I4 @ I6 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I4 ) ) )
     => ( ( ( groups8292507037921071086at_int @ X @ I6 )
          = one_one_int )
       => ( ! [I4: set_nat] :
              ( ( member_set_nat @ I4 @ I6 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I4 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups8292507037921071086at_int
                  @ ^ [I3: set_nat] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7539_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: nat > int,A: nat > int,B2: int,Delta: int] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ I6 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I4 ) ) )
     => ( ( ( groups3539618377306564664at_int @ X @ I6 )
          = one_one_int )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I6 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I4 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7540_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > int,A: int > int,B2: int,Delta: int] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I6 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I4 ) ) )
     => ( ( ( groups4538972089207619220nt_int @ X @ I6 )
          = one_one_int )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I4 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups4538972089207619220nt_int
                  @ ^ [I3: int] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7541_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: nat > real,A: nat > real,B2: real,Delta: real] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
     => ( ( ( groups6591440286371151544t_real @ X @ I6 )
          = one_one_real )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7542_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7543_arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arctan @ X )
        = ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_7544_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D4: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z6: int] :
                ( ( ord_less_eq_int @ D4 @ Z7 )
                & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_7545_Maclaurin__exp__lt,axiom,
    ! [X: real,N2: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [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
                  @ ^ [M: nat] : ( divide_divide_real @ ( power_power_real @ X @ M ) @ ( semiri2265585572941072030t_real @ M ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_7546_xor__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_7547_zdvd1__eq,axiom,
    ! [X: int] :
      ( ( dvd_dvd_int @ X @ one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_7548_zabs__less__one__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z3 ) @ one_one_int )
      = ( Z3 = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_7549_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_7550_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_7551_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_7552_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_7553_zdvd__antisym__abs,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( abs_abs_int @ A )
          = ( abs_abs_int @ B2 ) ) ) ) ).

% zdvd_antisym_abs
thf(fact_7554_abs__zmult__eq__1,axiom,
    ! [M2: int,N2: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M2 @ N2 ) )
        = one_one_int )
     => ( ( abs_abs_int @ M2 )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_7555_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M: int] :
          ? [N: int] :
            ( ( ord_less_eq_int @ M @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_7556_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M: int] :
          ? [N: int] :
            ( ( ord_less_int @ M @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_7557_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).

% zabs_def
thf(fact_7558_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_7559_zdvd__mult__cancel1,axiom,
    ! [M2: int,N2: int] :
      ( ( M2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M2 @ N2 ) @ M2 )
        = ( ( abs_abs_int @ N2 )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_7560_nat__intermed__int__val,axiom,
    ! [M2: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ( ord_less_eq_nat @ M2 @ I4 )
            & ( ord_less_nat @ I4 @ N2 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( ord_less_eq_int @ ( F @ M2 ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
           => ? [I4: nat] :
                ( ( ord_less_eq_nat @ M2 @ I4 )
                & ( ord_less_eq_nat @ I4 @ N2 )
                & ( ( F @ I4 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_7561_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I4: nat] :
              ( ( ord_less_eq_nat @ I4 @ N2 )
              & ( ( F @ I4 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_7562_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_7563_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I4 @ one_one_nat ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I4: nat] :
              ( ( ord_less_eq_nat @ I4 @ N2 )
              & ( ( F @ I4 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_7564_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_7565_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_7566_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_7567_Suc__0__xor__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_7568_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K2: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_7569_nat__int,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( semiri1314217659103216013at_int @ N2 ) )
      = N2 ) ).

% nat_int
thf(fact_7570_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_7571_nat__of__bool,axiom,
    ! [P: $o] :
      ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% nat_of_bool
thf(fact_7572_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_7573_nat__le__0,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ Z3 @ zero_zero_int )
     => ( ( nat2 @ Z3 )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_7574_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_7575_zless__nat__conj,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
      = ( ( ord_less_int @ zero_zero_int @ Z3 )
        & ( ord_less_int @ W2 @ Z3 ) ) ) ).

% zless_nat_conj
thf(fact_7576_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_7577_nat__zminus__int,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_7578_int__nat__eq,axiom,
    ! [Z3: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
          = Z3 ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_7579_zero__less__nat__eq,axiom,
    ! [Z3: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
      = ( ord_less_int @ zero_zero_int @ Z3 ) ) ).

% zero_less_nat_eq
thf(fact_7580_diff__nat__numeral,axiom,
    ! [V: num,V3: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).

% diff_nat_numeral
thf(fact_7581_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_7582_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_7583_dvd__nat__abs__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( dvd_dvd_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ).

% dvd_nat_abs_iff
thf(fact_7584_nat__abs__dvd__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
      = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_abs_dvd_iff
thf(fact_7585_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_7586_one__less__nat__eq,axiom,
    ! [Z3: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
      = ( ord_less_int @ one_one_int @ Z3 ) ) ).

% one_less_nat_eq
thf(fact_7587_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_7588_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_7589_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_7590_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_7591_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_7592_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_7593_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_7594_ex__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X7: nat] : ( P2 @ X7 ) )
    = ( ^ [P3: nat > $o] :
        ? [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
          & ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).

% ex_nat
thf(fact_7595_all__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ! [X7: nat] : ( P2 @ X7 ) )
    = ( ^ [P3: nat > $o] :
        ! [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
         => ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).

% all_nat
thf(fact_7596_eq__nat__nat__iff,axiom,
    ! [Z3: int,Z8: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
       => ( ( ( nat2 @ Z3 )
            = ( nat2 @ Z8 ) )
          = ( Z3 = Z8 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_7597_nat__mono__iff,axiom,
    ! [Z3: int,W2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
        = ( ord_less_int @ W2 @ Z3 ) ) ) ).

% nat_mono_iff
thf(fact_7598_zless__nat__eq__int__zless,axiom,
    ! [M2: nat,Z3: int] :
      ( ( ord_less_nat @ M2 @ ( nat2 @ Z3 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z3 ) ) ).

% zless_nat_eq_int_zless
thf(fact_7599_nat__le__iff,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_le_iff
thf(fact_7600_int__eq__iff,axiom,
    ! [M2: nat,Z3: int] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = Z3 )
      = ( ( M2
          = ( nat2 @ Z3 ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).

% int_eq_iff
thf(fact_7601_nat__0__le,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
        = Z3 ) ) ).

% nat_0_le
thf(fact_7602_nat__int__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
      = ( plus_plus_nat @ A @ B2 ) ) ).

% nat_int_add
thf(fact_7603_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z3: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z3 ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z3 ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_7604_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_7605_nat__less__eq__zless,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
        = ( ord_less_int @ W2 @ Z3 ) ) ) ).

% nat_less_eq_zless
thf(fact_7606_nat__le__eq__zle,axiom,
    ! [W2: int,Z3: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W2 )
        | ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
        = ( ord_less_eq_int @ W2 @ Z3 ) ) ) ).

% nat_le_eq_zle
thf(fact_7607_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_7608_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_7609_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N ) )
           => ( P @ N ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_7610_le__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).

% le_nat_iff
thf(fact_7611_nat__add__distrib,axiom,
    ! [Z3: int,Z8: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
       => ( ( nat2 @ ( plus_plus_int @ Z3 @ Z8 ) )
          = ( plus_plus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z8 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_7612_nat__mult__distrib,axiom,
    ! [Z3: int,Z8: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( nat2 @ ( times_times_int @ Z3 @ Z8 ) )
        = ( times_times_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z8 ) ) ) ) ).

% nat_mult_distrib
thf(fact_7613_Suc__as__int,axiom,
    ( suc
    = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_7614_forall__pos__mono__1,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D5: real,E: real] :
          ( ( ord_less_real @ D5 @ E )
         => ( ( P @ D5 )
           => ( P @ E ) ) )
     => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_7615_real__arch__inverse,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
      = ( ? [N: nat] :
            ( ( N != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E2 ) ) ) ) ).

% real_arch_inverse
thf(fact_7616_forall__pos__mono,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D5: real,E: real] :
          ( ( ord_less_real @ D5 @ E )
         => ( ( P @ D5 )
           => ( P @ E ) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono
thf(fact_7617_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_7618_nat__diff__distrib,axiom,
    ! [Z8: int,Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
     => ( ( ord_less_eq_int @ Z8 @ Z3 )
       => ( ( nat2 @ ( minus_minus_int @ Z3 @ Z8 ) )
          = ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z8 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_7619_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_7620_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_7621_nat__power__eq,axiom,
    ! [Z3: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( nat2 @ ( power_power_int @ Z3 @ N2 ) )
        = ( power_power_nat @ ( nat2 @ Z3 ) @ N2 ) ) ) ).

% nat_power_eq
thf(fact_7622_floor__eq3,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N2 ) ) ) ).

% floor_eq3
thf(fact_7623_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_7624_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_7625_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( suc @ ( nat2 @ Z3 ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_7626_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_7627_nat__mult__distrib__neg,axiom,
    ! [Z3: int,Z8: int] :
      ( ( ord_less_eq_int @ Z3 @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z3 @ Z8 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z8 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_7628_nat__abs__int__diff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ord_less_eq_nat @ A @ B2 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
          = ( minus_minus_nat @ B2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B2 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
          = ( minus_minus_nat @ A @ B2 ) ) ) ) ).

% nat_abs_int_diff
thf(fact_7629_floor__eq4,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N2 ) ) ) ).

% floor_eq4
thf(fact_7630_diff__nat__eq__if,axiom,
    ! [Z8: int,Z3: int] :
      ( ( ( ord_less_int @ Z8 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z8 ) )
          = ( nat2 @ Z3 ) ) )
      & ( ~ ( ord_less_int @ Z8 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z8 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z3 @ Z8 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z3 @ Z8 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_7631_nat__dvd__iff,axiom,
    ! [Z3: int,M2: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z3 ) @ M2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
         => ( dvd_dvd_int @ Z3 @ ( semiri1314217659103216013at_int @ M2 ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_7632_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_7633_Sum__Ico__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_7634_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_7635_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X6: nat > real] :
        ! [J2: nat] :
        ? [M9: nat] :
        ! [M: nat] :
          ( ( ord_less_eq_nat @ M9 @ M )
         => ! [N: nat] :
              ( ( ord_less_eq_nat @ M9 @ N )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M ) @ ( X6 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J2 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_7636_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).

% finite_atLeastLessThan
thf(fact_7637_atLeastLessThan__singleton,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
      = ( insert_nat @ M2 @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_7638_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_7639_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_7640_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_7641_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_7642_ex__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M: nat] :
            ( ( ord_less_nat @ M @ N2 )
            & ( P @ M ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X4 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_7643_all__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M: nat] :
            ( ( ord_less_nat @ M @ N2 )
           => ( P @ M ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X4 ) ) ) ) ).

% all_nat_less_eq
thf(fact_7644_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_7645_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_7646_atLeastLessThan0,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_7647_atLeast0__lessThan__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_7648_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N6 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_7649_atLeastLessThanSuc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) )
          = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_7650_prod__Suc__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_Suc_fact
thf(fact_7651_prod__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_fact
thf(fact_7652_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_7653_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_7654_Suc__0__or__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_or_eq
thf(fact_7655_or__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% or_Suc_0_eq
thf(fact_7656_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_7657_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_7658_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A: nat > nat,B2: nat > nat] :
      ( ! [I4: nat,J3: nat] :
          ( ( ord_less_eq_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ N2 )
           => ( ord_less_eq_nat @ ( A @ I4 ) @ ( A @ J3 ) ) ) )
     => ( ! [I4: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ N2 )
             => ( ord_less_eq_nat @ ( B2 @ J3 ) @ ( B2 @ I4 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N2
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B2 @ I3 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_7659_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_7660_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_7661_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_7662_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_7663_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size_gen(2)
thf(fact_7664_card__lessThan,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
      = U ) ).

% card_lessThan
thf(fact_7665_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_7666_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_7667_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_7668_card__Collect__le__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) )
      = ( suc @ N2 ) ) ).

% card_Collect_le_nat
thf(fact_7669_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_7670_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_7671_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_7672_push__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_Suc_0
thf(fact_7673_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_7674_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_7675_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_7676_bit__push__bit__iff__int,axiom,
    ! [M2: nat,K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M2 @ K ) @ N2 )
      = ( ( ord_less_eq_nat @ M2 @ N2 )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_7677_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ ( suc @ K2 ) @ M7 )
                & ( ord_less_nat @ K2 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_7678_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K2: nat] :
                  ( ( member_nat @ ( suc @ K2 ) @ M7 )
                  & ( ord_less_nat @ K2 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_7679_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_7680_bit__push__bit__iff__nat,axiom,
    ! [M2: nat,Q3: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M2 @ Q3 ) @ N2 )
      = ( ( ord_less_eq_nat @ M2 @ N2 )
        & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_7681_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_7682_less__eq__nat_Osimps_I2_J,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M2 ) @ N2 ) ) ).

% less_eq_nat.simps(2)
thf(fact_7683_max__Suc2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_max_nat @ M2 @ ( suc @ N2 ) )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ M4 @ N2 ) )
        @ M2 ) ) ).

% max_Suc2
thf(fact_7684_max__Suc1,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ M2 )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ N2 @ M4 ) )
        @ M2 ) ) ).

% max_Suc1
thf(fact_7685_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N6 ) @ N2 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_7686_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ S2 ) ) ).

% card_sum_le_nat_sum
thf(fact_7687_card__nth__roots,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z6: complex] :
                  ( ( power_power_complex @ Z6 @ N2 )
                  = C ) ) )
          = N2 ) ) ) ).

% card_nth_roots
thf(fact_7688_card__roots__unity__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = one_one_complex ) ) )
        = N2 ) ) ).

% card_roots_unity_eq
thf(fact_7689_diff__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K2: nat] : K2
        @ ( minus_minus_nat @ M2 @ N2 ) ) ) ).

% diff_Suc
thf(fact_7690_binomial__def,axiom,
    ( binomial
    = ( ^ [N: nat,K2: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K7: set_nat] :
                ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
                & ( ( finite_card_nat @ K7 )
                  = K2 ) ) ) ) ) ) ).

% binomial_def
thf(fact_7691_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_7692_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_7693_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K2: nat,M: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M @ K2 ) @ ( product_Pair_nat_nat @ M @ ( minus_minus_nat @ K2 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M @ ( suc @ K2 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_7694_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_7695_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [R3: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
          & ! [N7: nat] :
              ( ( ord_less_nat @ N7 @ ( finite_card_nat @ S2 ) )
             => ( member_nat @ ( R3 @ N7 ) @ S2 ) ) ) ) ).

% finite_enumerate
thf(fact_7696_root__powr__inverse,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N2 @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_7697_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_7698_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_7699_real__root__eq__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = ( root @ N2 @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_7700_drop__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_7701_drop__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_7702_real__root__eq__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_7703_real__root__less__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_7704_real__root__le__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_7705_real__root__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_7706_real__root__eq__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_7707_real__root__gt__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_7708_real__root__lt__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_7709_real__root__ge__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_7710_real__root__le__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_7711_real__root__gt__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_7712_real__root__lt__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_7713_real__root__ge__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_7714_real__root__le__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_7715_drop__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_7716_real__root__pow__pos2,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_7717_real__root__less__mono,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_7718_real__root__le__mono,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_7719_real__root__power,axiom,
    ! [N2: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N2 @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_7720_real__root__abs,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_abs
thf(fact_7721_sgn__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( sgn_sgn_real @ ( root @ N2 @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_7722_real__root__gt__zero,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_7723_real__root__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N6 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_7724_root__abs__power,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y @ N2 ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_7725_real__root__pos__pos,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_7726_real__root__strict__increasing,axiom,
    ! [N2: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_7727_real__root__decreasing,axiom,
    ! [N2: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N6 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_7728_real__root__pow__pos,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_7729_real__root__pos__unique,axiom,
    ! [N2: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N2 )
            = X )
         => ( ( root @ N2 @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_7730_real__root__power__cancel,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_7731_real__root__increasing,axiom,
    ! [N2: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_7732_root__sgn__power,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_7733_sgn__power__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X ) ) @ N2 ) )
        = X ) ) ).

% sgn_power_root
thf(fact_7734_ln__root,axiom,
    ! [N2: nat,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( ln_ln_real @ ( root @ N2 @ B2 ) )
          = ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% ln_root
thf(fact_7735_log__root,axiom,
    ! [N2: nat,A: real,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B2 @ ( root @ N2 @ A ) )
          = ( divide_divide_real @ ( log @ B2 @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_root
thf(fact_7736_log__base__root,axiom,
    ! [N2: nat,B2: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( log @ ( root @ N2 @ B2 ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B2 @ X ) ) ) ) ) ).

% log_base_root
thf(fact_7737_split__root,axiom,
    ! [P: real > $o,N2: nat,X: real] :
      ( ( P @ ( root @ N2 @ X ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ! [Y5: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N2 ) )
                = X )
             => ( P @ Y5 ) ) ) ) ) ).

% split_root
thf(fact_7738_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_7739_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => ( member_nat @ ( complete_Inf_Inf_nat @ K5 ) @ K5 ) ) ).

% Inf_nat_def1
thf(fact_7740_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_7741_Suc__funpow,axiom,
    ! [N2: nat] :
      ( ( compow_nat_nat @ N2 @ suc )
      = ( plus_plus_nat @ N2 ) ) ).

% Suc_funpow
thf(fact_7742_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_7743_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_7744_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_7745_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_7746_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_7747_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X4: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X4 )
    @ ^ [X4: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X4 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_7748_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X4: nat,Y5: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) )
          @ Xa2
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_7749_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( gcd_Gcd_nat @ M7 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_7750_int_Oabs__induct,axiom,
    ! [P: int > $o,X: int] :
      ( ! [Y3: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y3 ) )
     => ( P @ X ) ) ).

% int.abs_induct
thf(fact_7751_eq__Abs__Integ,axiom,
    ! [Z3: int] :
      ~ ! [X5: nat,Y3: nat] :
          ( Z3
         != ( abs_Integ @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) ) ).

% eq_Abs_Integ
thf(fact_7752_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( nat2 @ ( abs_Integ @ X ) )
      = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).

% nat.abs_eq
thf(fact_7753_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_7754_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_7755_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_7756_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_7757_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
        @ ^ [X4: nat,Y5: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) )
        @ Xa2
        @ X ) ) ).

% less_int.abs_eq
thf(fact_7758_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
        @ ^ [X4: nat,Y5: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) )
        @ Xa2
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_7759_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X4: nat,Y5: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) )
          @ Xa2
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_7760_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X4: nat,Y5: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) )
          @ Xa2
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_7761_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = ( inc @ ( num_of_nat @ N2 ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_7762_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_7763_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
        = N2 ) ) ).

% numeral_num_of_nat
thf(fact_7764_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ one_one_nat )
     => ( ( num_of_nat @ N2 )
        = one ) ) ).

% num_of_nat_One
thf(fact_7765_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
        = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).

% num_of_nat_double
thf(fact_7766_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( plus_plus_nat @ M2 @ N2 ) )
          = ( plus_plus_num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N2 ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_7767_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_7768_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Xa3: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y5: nat,Z6: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa3 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_7769_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Xa3: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y5: nat,Z6: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa3 ) ) ) ) ).

% less_int.rep_eq
thf(fact_7770_nat_Orep__eq,axiom,
    ( nat2
    = ( ^ [X4: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X4 ) ) ) ) ).

% nat.rep_eq
thf(fact_7771_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_7772_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) ) ) ) ).

% uminus_int_def
thf(fact_7773_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M: nat,N: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M @ N ) ) @ M ) ) ) ).

% prod_encode_def
thf(fact_7774_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_7775_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_greaterThanAtMost
thf(fact_7776_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_7777_le__prod__encode__1,axiom,
    ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).

% le_prod_encode_1
thf(fact_7778_le__prod__encode__2,axiom,
    ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).

% le_prod_encode_2
thf(fact_7779_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M2: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ K ) @ M2 ) ) ).

% prod_encode_prod_decode_aux
thf(fact_7780_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y5: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_7781_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y5: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_7782_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y5: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_7783_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ 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
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_7784_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_7785_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_7786_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_7787_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_7788_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_7789_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_7790_image__Suc__lessThan,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).

% image_Suc_lessThan
thf(fact_7791_image__Suc__atMost,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).

% image_Suc_atMost
thf(fact_7792_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_7793_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_7794_lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_7795_atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_7796_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_7797_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y5: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ) ).

% less_int_def
thf(fact_7798_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
        @ ^ [X4: nat,Y5: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_7799_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S6: set_int] :
        ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_7800_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S6: set_int] :
        ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_7801_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_7802_image__int__atLeastAtMost,axiom,
    ! [A: nat,B2: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% image_int_atLeastAtMost
thf(fact_7803_image__int__atLeastLessThan,axiom,
    ! [A: nat,B2: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B2 ) )
      = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% image_int_atLeastLessThan
thf(fact_7804_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X4: int] : ( plus_plus_int @ X4 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_7805_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_7806_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_7807_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_7808_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_7809_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_7810_infinite__UNIV__int,axiom,
    ~ ( finite_finite_int @ top_top_set_int ) ).

% infinite_UNIV_int
thf(fact_7811_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_7812_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_7813_int__in__range__abs,axiom,
    ! [N2: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).

% int_in_range_abs
thf(fact_7814_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_7815_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M: nat] : ( modulo_modulo_nat @ M @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_7816_root__def,axiom,
    ( root
    = ( ^ [N: nat,X4: real] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N ) )
            @ X4 ) ) ) ) ).

% root_def
thf(fact_7817_DERIV__real__root__generic,axiom,
    ! [N2: nat,X: real,D6: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D6
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D6
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
               => ( D6
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_7818_DERIV__even__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_7819_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X5 @ N ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X4: real] :
                ( suminf_real
                @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X4 @ ( suc @ N ) ) ) )
            @ ( suminf_real
              @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X0 @ N ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_7820_DERIV__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_7821_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ 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
                  @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ X @ M ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_7822_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M3: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ 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
                @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ X @ M ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_7823_DERIV__odd__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_7824_Maclaurin__minus,axiom,
    ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M3: nat,T6: real] :
                ( ( ( ord_less_nat @ M3 @ N2 )
                  & ( ord_less_eq_real @ H2 @ T6 )
                  & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ H2 @ T6 )
                & ( ord_less_real @ T6 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ H2 @ M ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_7825_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ zero_zero_real @ T6 )
                & ( ord_less_eq_real @ T6 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ? [T6: real] :
              ( ( ord_less_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ H2 @ M ) )
                    @ ( set_ord_lessThan_nat @ N2 ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_7826_Maclaurin,axiom,
    ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M3: nat,T6: real] :
                ( ( ( ord_less_nat @ M3 @ N2 )
                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
                  & ( ord_less_eq_real @ T6 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ T6 )
                & ( ord_less_real @ T6 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ H2 @ M ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_7827_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( X != zero_zero_real )
         => ( ! [M3: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ 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
                      @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ X @ M ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_7828_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,T6: real] :
            ( ( ( ord_less_nat @ M3 @ N2 )
              & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ 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
                  @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ X @ M ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_7829_Taylor,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B2 )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B2 )
                 => ( ( 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
                              @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ C ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M ) )
                              @ ( set_ord_lessThan_nat @ N2 ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_7830_Taylor__up,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B2 )
             => ? [T6: real] :
                  ( ( ord_less_real @ C @ T6 )
                  & ( ord_less_real @ T6 @ B2 )
                  & ( ( F @ B2 )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ C ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ M ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_7831_Taylor__down,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B2 )
             => ? [T6: real] :
                  ( ( ord_less_real @ A @ T6 )
                  & ( ord_less_real @ T6 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M @ C ) @ ( semiri2265585572941072030t_real @ M ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_7832_Maclaurin__lemma2,axiom,
    ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B: real] :
      ( ! [M3: nat,T6: real] :
          ( ( ( ord_less_nat @ M3 @ N2 )
            & ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
     => ( ( N2
          = ( suc @ K ) )
       => ! [M5: nat,T7: real] :
            ( ( ( ord_less_nat @ M5 @ N2 )
              & ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M5 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M5 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U2 @ P6 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M5 ) ) )
                    @ ( times_times_real @ B @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M5 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M5 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M5 ) @ T7 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M5 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T7 @ P6 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M5 ) ) ) )
                  @ ( times_times_real @ B @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N2 @ ( suc @ M5 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M5 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_7833_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_7834_DERIV__pow,axiom,
    ! [N2: nat,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X4: real] : ( power_power_real @ X4 @ N2 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S ) ) ).

% DERIV_pow
thf(fact_7835_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M7 )
         => ( ( gcd_Gcd_nat @ M7 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M: nat] :
                      ( collect_nat
                      @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) )
                  @ M7 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_7836_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N2 ) ) )
        = N2 ) ) ).

% Max_divisors_self_nat
thf(fact_7837_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_7838_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X6: set_nat] : ( if_nat @ ( X6 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X6 ) ) ) ) ).

% Sup_nat_def
thf(fact_7839_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N ) @ M ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_7840_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 )
         => ! [N7: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_7841_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 ) )
         => ! [N7: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_7842_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_7843_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_7844_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_7845_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_7846_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X4: nat] : ( times_times_nat @ X4 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_7847_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_7848_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
         => ( ( filterlim_nat_real
              @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N7: nat] : ( ord_less_eq_real @ ( F @ N7 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N7: nat] : ( ord_less_eq_real @ L4 @ ( G @ N7 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_7849_LIMSEQ__inverse__zero,axiom,
    ! [X8: nat > real] :
      ( ! [R3: real] :
        ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_real @ R3 @ ( X8 @ N3 ) ) )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( inverse_inverse_real @ ( X8 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_7850_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_7851_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_7852_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
       => ( ! [E: real] :
              ( ( ord_less_real @ zero_zero_real @ E )
             => ? [N7: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N7 ) @ E ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_7853_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_7854_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_7855_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ) ).

% summable
thf(fact_7856_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_7857_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_7858_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_7859_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N7: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N7: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_7860_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
          @ ^ [N: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_7861_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_7862_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N: nat] : ( P @ ( plus_plus_nat @ N @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_7863_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X5: nat] :
          ( ( ord_less_eq_nat @ C @ X5 )
         => ( P @ X5 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_7864_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N5: nat] :
          ! [N: nat] :
            ( ( ord_less_eq_nat @ N5 @ N )
           => ( P @ N ) ) ) ) ).

% eventually_sequentially
thf(fact_7865_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F3 @ at_top_nat )
      = ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F3 ) ) ) ).

% le_sequentially
thf(fact_7866_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_7867_filterlim__pow__at__bot__even,axiom,
    ! [N2: nat,F: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
         => ( filterlim_real_real
            @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N2 )
            @ at_top_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_7868_filterlim__pow__at__bot__odd,axiom,
    ! [N2: nat,F: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
         => ( filterlim_real_real
            @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N2 )
            @ at_bot_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_7869_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_7870_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_7871_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_7872_mono__times__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).

% mono_times_nat
thf(fact_7873_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_7874_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_7875_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ M ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_7876_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_7877_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_7878_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_7879_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_7880_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_7881_range__abs__Nats,axiom,
    ( ( image_int_int @ abs_abs_int @ top_top_set_int )
    = semiring_1_Nats_int ) ).

% range_abs_Nats
thf(fact_7882_inj__sgn__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( inj_on_real_real
        @ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N2 ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_7883_measure__function__int,axiom,
    fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).

% measure_function_int
thf(fact_7884_inj__Suc,axiom,
    ! [N6: set_nat] : ( inj_on_nat_nat @ suc @ N6 ) ).

% inj_Suc
thf(fact_7885_inj__on__diff__nat,axiom,
    ! [N6: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ N6 )
         => ( ord_less_eq_nat @ K @ N3 ) )
     => ( inj_on_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
        @ N6 ) ) ).

% inj_on_diff_nat
thf(fact_7886_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_7887_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M: nat,N: nat] :
            ( N
            = ( suc @ M ) ) ) ) ) ).

% pred_nat_def
thf(fact_7888_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,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                    & ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I3: nat] :
                            ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                            & ! [X4: nat] :
                                ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
                                 => ( ( ord_less_nat @ Mi3 @ X4 )
                                    & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_7889_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,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( 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 @ TreeList3 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X4: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                              & ! [X4: nat] :
                                  ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
                                   => ( ( ord_less_nat @ Mi3 @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_7890_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,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( 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 @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_7891_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
      = ( ( Deg = Deg3 )
        & ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
            & ! [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                    & ! [X4: nat] :
                        ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_7892_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,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( 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 @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_7893_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,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( 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 @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_7894_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,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                        & ! [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I3: nat] :
                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X4: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                & ! [X4: nat] :
                                    ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X4 )
                                     => ( ( ord_less_nat @ Mi3 @ X4 )
                                        & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_7895_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_7896_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
thf(fact_7897_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_7898_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I3: int,N: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N ) ) )
          & ( N != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_7899_less__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N2 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% less_eq
thf(fact_7900_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N2 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% pred_nat_trancl_eq_le
thf(fact_7901_bdd__above__nat,axiom,
    condit2214826472909112428ve_nat = finite_finite_nat ).

% bdd_above_nat
thf(fact_7902_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_7903_take__bit__num__simps_I2_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_7904_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M2: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M2 ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ N2 @ M2 ) ) ) ).

% take_bit_num_simps(3)
thf(fact_7905_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M2: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M2 ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_7906_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_7907_min__Suc__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ M2 @ N2 ) ) ) ).

% min_Suc_Suc
thf(fact_7908_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_7909_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_7910_min__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% min_numeral_Suc
thf(fact_7911_min__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_7912_min__diff,axiom,
    ! [M2: nat,I: nat,N2: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I ) @ ( minus_minus_nat @ N2 @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M2 @ N2 ) @ I ) ) ).

% min_diff
thf(fact_7913_nat__mult__min__left,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M2 @ N2 ) @ Q3 )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ Q3 ) @ ( times_times_nat @ N2 @ Q3 ) ) ) ).

% nat_mult_min_left
thf(fact_7914_nat__mult__min__right,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( times_times_nat @ M2 @ ( ord_min_nat @ N2 @ Q3 ) )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ N2 ) @ ( times_times_nat @ M2 @ Q3 ) ) ) ).

% nat_mult_min_right
thf(fact_7915_min__Suc2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_min_nat @ M2 @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ M4 @ N2 ) )
        @ M2 ) ) ).

% min_Suc2
thf(fact_7916_min__Suc1,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ M2 )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ N2 @ M4 ) )
        @ M2 ) ) ).

% min_Suc1
thf(fact_7917_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_7918_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
thf(fact_7919_bij__betw__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( bij_betw_nat_complex
        @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 )
        @ ( collect_complex
          @ ^ [Z6: complex] :
              ( ( power_power_complex @ Z6 @ N2 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_7920_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_7921_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_7922_bij__betw__Suc,axiom,
    ! [M7: set_nat,N6: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N6 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N6 ) ) ).

% bij_betw_Suc
thf(fact_7923_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N2 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_7924_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_7925_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_7926_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X5: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X5 @ Xs2 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_7927_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_7928_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_7929_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I3: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_7930_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_7931_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_7932_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_7933_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_7934_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_7935_upto__single,axiom,
    ! [I: int] :
      ( ( upto @ I @ I )
      = ( cons_int @ I @ nil_int ) ) ).

% upto_single
thf(fact_7936_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_7937_length__upto,axiom,
    ! [I: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_7938_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_7939_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_7940_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
          = ( 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 @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_7941_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( 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 @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_7942_upto__code,axiom,
    ( upto
    = ( ^ [I3: int,J2: int] : ( upto_aux @ I3 @ J2 @ nil_int ) ) ) ).

% upto_code
thf(fact_7943_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I3: int,J2: int] : ( append_int @ ( upto @ I3 @ J2 ) ) ) ) ).

% upto_aux_def
thf(fact_7944_distinct__upto,axiom,
    ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).

% distinct_upto
thf(fact_7945_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ I3 @ J2 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_7946_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_7947_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_7948_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_7949_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_7950_upto_Osimps,axiom,
    ( upto
    = ( ^ [I3: int,J2: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J2 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_7951_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_7952_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_7953_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_7954_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_7955_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_7956_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 ) ) )
         => ~ ! [X5: nat,Xs2: list_nat] :
                ( ( X
                  = ( cons_nat @ X5 @ Xs2 ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs2 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs2 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_7957_upt__rec__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_7958_remdups__upt,axiom,
    ! [M2: nat,N2: nat] :
      ( ( remdups_nat @ ( upt @ M2 @ N2 ) )
      = ( upt @ M2 @ N2 ) ) ).

% remdups_upt
thf(fact_7959_tl__upt,axiom,
    ! [M2: nat,N2: nat] :
      ( ( tl_nat @ ( upt @ M2 @ N2 ) )
      = ( upt @ ( suc @ M2 ) @ N2 ) ) ).

% tl_upt
thf(fact_7960_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_7961_drop__upt,axiom,
    ! [M2: nat,I: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_7962_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_7963_take__upt,axiom,
    ! [I: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M2 ) @ N2 )
     => ( ( take_nat @ M2 @ ( upt @ I @ N2 ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M2 ) ) ) ) ).

% take_upt
thf(fact_7964_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_7965_sorted__list__of__set__range,axiom,
    ! [M2: nat,N2: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
      = ( upt @ M2 @ N2 ) ) ).

% sorted_list_of_set_range
thf(fact_7966_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_7967_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_7968_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_7969_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N2: nat,Ns: list_nat,Q3: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N2 @ Ns ) )
        = ( upt @ M2 @ Q3 ) )
      = ( ( cons_nat @ N2 @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_7970_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N: nat,M: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ ( suc @ M ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_7971_distinct__upt,axiom,
    ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).

% distinct_upt
thf(fact_7972_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N: nat,M: nat] : ( set_nat2 @ ( upt @ N @ ( suc @ M ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_7973_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I3: nat,J2: nat] : ( set_nat2 @ ( upt @ I3 @ J2 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_7974_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N ) ) ) ) ).

% atLeast_upt
thf(fact_7975_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N: nat,M: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ M ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_7976_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).

% atMost_upto
thf(fact_7977_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_7978_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_7979_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_7980_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J2 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J2 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_7981_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_7982_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_7983_map__Suc__upt,axiom,
    ! [M2: nat,N2: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M2 @ N2 ) )
      = ( upt @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).

% map_Suc_upt
thf(fact_7984_map__add__upt,axiom,
    ! [N2: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N2 )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% map_add_upt
thf(fact_7985_map__decr__upt,axiom,
    ! [M2: nat,N2: nat] :
      ( ( map_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( upt @ M2 @ N2 ) ) ).

% map_decr_upt
thf(fact_7986_sum__list__upt,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M2 @ N2 ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ) ).

% sum_list_upt
thf(fact_7987_card__length__sum__list__rec,axiom,
    ! [M2: nat,N6: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L2: list_nat] :
                ( ( ( size_size_list_nat @ L2 )
                  = M2 )
                & ( ( groups4561878855575611511st_nat @ L2 )
                  = N6 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = ( minus_minus_nat @ M2 @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L2 )
                    = N6 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = M2 )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
                    = N6 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_7988_card__length__sum__list,axiom,
    ! [M2: nat,N6: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M2 )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N6 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N6 @ M2 ) @ one_one_nat ) @ N6 ) ) ).

% card_length_sum_list
thf(fact_7989_sorted__wrt__upt,axiom,
    ! [M2: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M2 @ N2 ) ) ).

% sorted_wrt_upt
thf(fact_7990_sorted__upt,axiom,
    ! [M2: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N2 ) ) ).

% sorted_upt
thf(fact_7991_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_7992_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_7993_sorted__upto,axiom,
    ! [M2: int,N2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M2 @ N2 ) ) ).

% sorted_upto
thf(fact_7994_pairs__le__eq__Sigma,axiom,
    ! [M2: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I3: nat,J2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J2 ) @ M2 ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M2 )
        @ ^ [R4: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M2 @ R4 ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_7995_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_7996_Restr__natLeq,axiom,
    ! [N2: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N2 ) )
          @ ^ [Uu3: nat] :
              ( collect_nat
              @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N2 ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ N2 )
              & ( ord_less_nat @ Y5 @ N2 )
              & ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_7997_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_7998_Restr__natLeq2,axiom,
    ! [N2: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
          @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ N2 )
              & ( ord_less_nat @ Y5 @ N2 )
              & ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_7999_natLeq__underS__less,axiom,
    ! [N2: nat] :
      ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
      = ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N2 ) ) ) ).

% natLeq_underS_less
thf(fact_8000_sort__upt,axiom,
    ! [M2: nat,N2: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X4: nat] : X4
        @ ( upt @ M2 @ N2 ) )
      = ( upt @ M2 @ N2 ) ) ).

% sort_upt
thf(fact_8001_sort__upto,axiom,
    ! [I: int,J: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X4: int] : X4
        @ ( upto @ I @ J ) )
      = ( upto @ I @ J ) ) ).

% sort_upto
thf(fact_8002_Field__natLeq__on,axiom,
    ! [N2: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y5: nat] :
                ( ( ord_less_nat @ X4 @ N2 )
                & ( ord_less_nat @ Y5 @ N2 )
                & ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) )
      = ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N2 ) ) ) ).

% Field_natLeq_on
thf(fact_8003_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_8004_wf__int__ge__less__than2,axiom,
    ! [D: int] : ( wf_int @ ( int_ge_less_than2 @ D ) ) ).

% wf_int_ge_less_than2
thf(fact_8005_wf__int__ge__less__than,axiom,
    ! [D: int] : ( wf_int @ ( int_ge_less_than @ D ) ) ).

% wf_int_ge_less_than
thf(fact_8006_infinite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ? [R3: nat > nat] :
          ( ( order_5726023648592871131at_nat @ R3 )
          & ! [N7: nat] : ( member_nat @ ( R3 @ N7 ) @ S2 ) ) ) ).

% infinite_enumerate
thf(fact_8007_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( order_5726023648592871131at_nat @ F )
     => ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).

% strict_mono_imp_increasing
thf(fact_8008_enumerate__Ex,axiom,
    ! [S2: set_nat,S: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( ( member_nat @ S @ S2 )
       => ? [N3: nat] :
            ( ( infini8530281810654367211te_nat @ S2 @ N3 )
            = S ) ) ) ).

% enumerate_Ex
thf(fact_8009_le__enumerate,axiom,
    ! [S2: set_nat,N2: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S2 @ N2 ) ) ) ).

% le_enumerate
thf(fact_8010_strict__mono__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( order_5726023648592871131at_nat @ ( infini8530281810654367211te_nat @ S2 ) ) ) ).

% strict_mono_enumerate
thf(fact_8011_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_8012_finite__le__enumerate,axiom,
    ! [S2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S2 ) )
       => ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S2 @ N2 ) ) ) ) ).

% finite_le_enumerate
thf(fact_8013_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_8014_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_8015_Least__Suc2,axiom,
    ! [P: nat > $o,N2: nat,Q: nat > $o,M2: nat] :
      ( ( P @ N2 )
     => ( ( Q @ M2 )
       => ( ~ ( P @ zero_zero_nat )
         => ( ! [K3: nat] :
                ( ( P @ ( suc @ K3 ) )
                = ( Q @ K3 ) )
           => ( ( ord_Least_nat @ P )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_8016_Least__Suc,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M: nat] : ( P @ ( suc @ M ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_8017_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_8018_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_8019_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F3 ) )
      = ( finite_finite_nat @ F3 ) ) ).

% finite_vimage_Suc_iff
thf(fact_8020_vimage__Suc__insert__Suc,axiom,
    ! [N2: nat,A2: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N2 ) @ A2 ) )
      = ( insert_nat @ N2 @ ( vimage_nat_nat @ suc @ A2 ) ) ) ).

% vimage_Suc_insert_Suc
thf(fact_8021_set__decode__div__2,axiom,
    ! [X: nat] :
      ( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).

% set_decode_div_2
thf(fact_8022_set__encode__vimage__Suc,axiom,
    ! [A2: set_nat] :
      ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A2 ) )
      = ( divide_divide_nat @ ( nat_set_encode @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% set_encode_vimage_Suc
thf(fact_8023_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_8024_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_8025_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B2: nat] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_8026_natLeq__on__wo__rel,axiom,
    ! [N2: nat] :
      ( bNF_We3818239936649020644el_nat
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ N2 )
              & ( ord_less_nat @ Y5 @ N2 )
              & ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).

% natLeq_on_wo_rel
thf(fact_8027_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_8028_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_8029_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_8030_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_8031_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X4: nat,Y5: nat] : ( if_Pro3027730157355071871nt_int @ ( Y5 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X4 @ Y5 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X4 @ Y5 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X4 @ Y5 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y5 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_8032_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_8033_pair__lessI2,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_8034_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z3 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z3 ) ) ).

% pair_less_iff1
thf(fact_8035_pair__lessI1,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_8036_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_8037_pair__leqI2,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_8038_pair__leqI1,axiom,
    ! [A: nat,B2: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_8039_gcd__nat_Oordering__top__axioms,axiom,
    ( ordering_top_nat @ dvd_dvd_nat
    @ ^ [M: nat,N: nat] :
        ( ( dvd_dvd_nat @ M @ N )
        & ( M != N ) )
    @ zero_zero_nat ) ).

% gcd_nat.ordering_top_axioms
thf(fact_8040_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X4: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X4 )
    @ ^ [X4: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X4 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_8041_less__eq__enat__def,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [M: extended_enat] :
          ( extended_case_enat_o
          @ ^ [N1: nat] :
              ( extended_case_enat_o
              @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
              @ $false
              @ M )
          @ $true ) ) ) ).

% less_eq_enat_def
thf(fact_8042_less__enat__def,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [M: extended_enat,N: extended_enat] :
          ( extended_case_enat_o
          @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N )
          @ $false
          @ M ) ) ) ).

% less_enat_def
thf(fact_8043_prod__decode__triangle__add,axiom,
    ! [K: nat,M2: nat] :
      ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M2 ) )
      = ( nat_prod_decode_aux @ K @ M2 ) ) ).

% prod_decode_triangle_add
thf(fact_8044_prod__decode__def,axiom,
    ( nat_prod_decode
    = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).

% prod_decode_def
thf(fact_8045_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 ) )
       => ( ! [N3: nat] :
              ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) )
             => ( ! [X2: nat,Y6: nat] :
                    ( ( ( product_Pair_nat_nat @ X2 @ Y6 )
                      = ( nat_prod_decode @ N3 ) )
                   => ( P @ Y6 ) )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_8046_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N3: nat] :
              ( ( X
                = ( suc @ N3 ) )
             => ( Y
               != ( produc2761476792215241774st_nat
                  @ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
                  @ ( nat_prod_decode @ N3 ) ) ) ) ) ) ).

% list_decode.elims
thf(fact_8047_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_8048_list__decode_Osimps_I1_J,axiom,
    ( ( nat_list_decode @ zero_zero_nat )
    = nil_nat ) ).

% list_decode.simps(1)
thf(fact_8049_list__decode_Opsimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) )
     => ( ( nat_list_decode @ ( suc @ N2 ) )
        = ( produc2761476792215241774st_nat
          @ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
          @ ( nat_prod_decode @ N2 ) ) ) ) ).

% list_decode.psimps(2)
thf(fact_8050_list__decode_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( nat_list_decode @ ( suc @ N2 ) )
      = ( produc2761476792215241774st_nat
        @ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
        @ ( nat_prod_decode @ N2 ) ) ) ).

% list_decode.simps(2)
thf(fact_8051_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 ) ) )
         => ~ ! [N3: nat] :
                ( ( X
                  = ( suc @ N3 ) )
               => ( ( Y
                    = ( produc2761476792215241774st_nat
                      @ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
                      @ ( nat_prod_decode @ N3 ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_8052_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_8053_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_8054_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_8055_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_8056_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_8057_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_8058_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( ( member_real @ X @ field_5140801741446780682s_real )
     => ~ ! [M3: nat,N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( ( ( abs_abs_real @ X )
                = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
             => ~ ( algebr934650988132801477me_nat @ M3 @ N3 ) ) ) ) ).

% Rats_abs_nat_div_natE
thf(fact_8059_coprime__Suc__left__nat,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ ( suc @ N2 ) @ N2 ) ).

% coprime_Suc_left_nat
thf(fact_8060_coprime__Suc__right__nat,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ N2 @ ( suc @ N2 ) ) ).

% coprime_Suc_right_nat
thf(fact_8061_coprime__Suc__0__left,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N2 ) ).

% coprime_Suc_0_left
thf(fact_8062_coprime__Suc__0__right,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ N2 @ ( suc @ zero_zero_nat ) ) ).

% coprime_Suc_0_right
thf(fact_8063_eventually__prod__sequentially,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N5: nat] :
          ! [M: nat] :
            ( ( ord_less_eq_nat @ N5 @ M )
           => ! [N: nat] :
                ( ( ord_less_eq_nat @ N5 @ N )
               => ( P @ ( product_Pair_nat_nat @ N @ M ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_8064_coprime__diff__one__left__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ N2 ) ) ).

% coprime_diff_one_left_nat
thf(fact_8065_coprime__diff__one__right__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( algebr934650988132801477me_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_8066_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_8067_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_8068_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y5: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_8069_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y5: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_8070_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_8071_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_8072_nat_Otransfer,axiom,
    ( bNF_re4555766996558763186at_nat @ pcr_int
    @ ^ [Y4: nat,Z2: nat] : Y4 = Z2
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ nat2 ) ).

% nat.transfer
thf(fact_8073_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y4: nat,Z2: nat] : Y4 = Z2
    @ pcr_int
    @ ^ [N: nat] : ( product_Pair_nat_nat @ N @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_8074_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_8075_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
      = ( ( plus_plus_nat @ X @ V )
        = ( plus_plus_nat @ U @ Y ) ) ) ).

% intrel_iff
thf(fact_8076_nat_Orsp,axiom,
    ( bNF_re8246922863344978751at_nat @ intrel
    @ ^ [Y4: nat,Z2: nat] : Y4 = Z2
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).

% nat.rsp
thf(fact_8077_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) ) ) ).

% uminus_int.rsp
thf(fact_8078_int_Oabs__eq__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( abs_Integ @ X )
        = ( abs_Integ @ Y ) )
      = ( intrel @ X @ Y ) ) ).

% int.abs_eq_iff
thf(fact_8079_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_8080_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_8081_intrel__def,axiom,
    ( intrel
    = ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y5: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] :
              ( ( plus_plus_nat @ X4 @ V4 )
              = ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ).

% intrel_def
thf(fact_8082_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y5: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y5: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ).

% less_int.rsp
thf(fact_8083_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y5: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y5: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_8084_int_Orel__eq__transfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
    @ intrel
    @ ^ [Y4: int,Z2: int] : Y4 = Z2 ) ).

% int.rel_eq_transfer
thf(fact_8085_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_8086_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_8087_int_Obi__total,axiom,
    bi_tot896582865486249351at_int @ pcr_int ).

% int.bi_total
thf(fact_8088_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_8089_elimnum,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        = ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).

% elimnum
thf(fact_8090_idiff__enat__0__right,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N2 @ ( extended_enat2 @ zero_zero_nat ) )
      = N2 ) ).

% idiff_enat_0_right
thf(fact_8091_idiff__enat__0,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N2 )
      = ( extended_enat2 @ zero_zero_nat ) ) ).

% idiff_enat_0
thf(fact_8092_plus__enat__simps_I1_J,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
      = ( extended_enat2 @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% plus_enat_simps(1)
thf(fact_8093_enat__ord__simps_I2_J,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% enat_ord_simps(2)
thf(fact_8094_enat__ord__simps_I1_J,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% enat_ord_simps(1)
thf(fact_8095_numeral__less__enat__iff,axiom,
    ! [M2: num,N2: nat] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ N2 ) ) ).

% numeral_less_enat_iff
thf(fact_8096_numeral__le__enat__iff,axiom,
    ! [M2: num,N2: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ N2 ) ) ).

% numeral_le_enat_iff
thf(fact_8097_Suc__ile__eq,axiom,
    ! [M2: nat,N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M2 ) ) @ N2 )
      = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M2 ) @ N2 ) ) ).

% Suc_ile_eq
thf(fact_8098_finite__enat__bounded,axiom,
    ! [A2: set_Extended_enat,N2: nat] :
      ( ! [Y3: extended_enat] :
          ( ( member_Extended_enat @ Y3 @ A2 )
         => ( ord_le2932123472753598470d_enat @ Y3 @ ( extended_enat2 @ N2 ) ) )
     => ( finite4001608067531595151d_enat @ A2 ) ) ).

% finite_enat_bounded
thf(fact_8099_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N2: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( extended_enat2 @ N2 ) )
      = ( ? [Y7: nat,X9: nat] :
            ( ( X
              = ( extended_enat2 @ X9 ) )
            & ( Y
              = ( extended_enat2 @ Y7 ) )
            & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y7 ) @ N2 ) ) ) ) ).

% iadd_le_enat_iff
thf(fact_8100_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat
        = ( extended_enat2 @ X ) )
      = ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
thf(fact_8101_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( ( extended_enat2 @ X )
        = zero_z5237406670263579293d_enat )
      = ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
thf(fact_8102_zero__enat__def,axiom,
    ( zero_z5237406670263579293d_enat
    = ( extended_enat2 @ zero_zero_nat ) ) ).

% zero_enat_def
thf(fact_8103_less__enatE,axiom,
    ! [N2: extended_enat,M2: nat] :
      ( ( ord_le72135733267957522d_enat @ N2 @ ( extended_enat2 @ M2 ) )
     => ~ ! [K3: nat] :
            ( ( N2
              = ( extended_enat2 @ K3 ) )
           => ~ ( ord_less_nat @ K3 @ M2 ) ) ) ).

% less_enatE
thf(fact_8104_elimcomplete,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
        = ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).

% elimcomplete
thf(fact_8105_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_8106_times__enat__simps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N2 ) )
          = zero_z5237406670263579293d_enat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N2 ) )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
thf(fact_8107_Inf__enat__def,axiom,
    ( comple2295165028678016749d_enat
    = ( ^ [A5: set_Extended_enat] :
          ( if_Extended_enat @ ( A5 = bot_bo7653980558646680370d_enat ) @ extend5688581933313929465d_enat
          @ ( ord_Le1955565732374568822d_enat
            @ ^ [X4: extended_enat] : ( member_Extended_enat @ X4 @ A5 ) ) ) ) ) ).

% Inf_enat_def
thf(fact_8108_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_8109_Sup__enat__def,axiom,
    ( comple4398354569131411667d_enat
    = ( ^ [A5: set_Extended_enat] : ( if_Extended_enat @ ( A5 = bot_bo7653980558646680370d_enat ) @ zero_z5237406670263579293d_enat @ ( if_Extended_enat @ ( finite4001608067531595151d_enat @ A5 ) @ ( lattic921264341876707157d_enat @ A5 ) @ extend5688581933313929465d_enat ) ) ) ) ).

% Sup_enat_def
thf(fact_8110_times__enat__def,axiom,
    ( times_7803423173614009249d_enat
    = ( ^ [M: extended_enat,N: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P6: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P6 ) )
              @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
              @ N )
          @ ( if_Extended_enat @ ( N = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
          @ M ) ) ) ).

% times_enat_def
thf(fact_8111_plus__enat__def,axiom,
    ( plus_p3455044024723400733d_enat
    = ( ^ [M: extended_enat,N: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P6: nat] : ( extended_enat2 @ ( plus_plus_nat @ O @ P6 ) )
              @ extend5688581933313929465d_enat
              @ N )
          @ extend5688581933313929465d_enat
          @ M ) ) ) ).

% plus_enat_def
thf(fact_8112_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_8113_eSuc__def,axiom,
    ( extended_eSuc
    = ( extend3600170679010898289d_enat
      @ ^ [N: nat] : ( extended_enat2 @ ( suc @ N ) )
      @ extend5688581933313929465d_enat ) ) ).

% eSuc_def
thf(fact_8114_enat__eSuc__iff,axiom,
    ! [Y: nat,X: extended_enat] :
      ( ( ( extended_enat2 @ Y )
        = ( extended_eSuc @ X ) )
      = ( ? [N: nat] :
            ( ( Y
              = ( suc @ N ) )
            & ( ( extended_enat2 @ N )
              = X ) ) ) ) ).

% enat_eSuc_iff
thf(fact_8115_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y: nat] :
      ( ( ( extended_eSuc @ X )
        = ( extended_enat2 @ Y ) )
      = ( ? [N: nat] :
            ( ( Y
              = ( suc @ N ) )
            & ( X
              = ( extended_enat2 @ N ) ) ) ) ) ).

% eSuc_enat_iff
thf(fact_8116_eSuc__enat,axiom,
    ! [N2: nat] :
      ( ( extended_eSuc @ ( extended_enat2 @ N2 ) )
      = ( extended_enat2 @ ( suc @ N2 ) ) ) ).

% eSuc_enat
thf(fact_8117_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_8118_natLeq__on__well__order__on,axiom,
    ! [N2: nat] :
      ( order_2888998067076097458on_nat
      @ ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N2 ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ N2 )
              & ( ord_less_nat @ Y5 @ N2 )
              & ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).

% natLeq_on_well_order_on
thf(fact_8119_natLeq__on__Well__order,axiom,
    ! [N2: nat] :
      ( order_2888998067076097458on_nat
      @ ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y5: nat] :
                ( ( ord_less_nat @ X4 @ N2 )
                & ( ord_less_nat @ Y5 @ N2 )
                & ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ N2 )
              & ( ord_less_nat @ Y5 @ N2 )
              & ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).

% natLeq_on_Well_order

% Helper facts (29)
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__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_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__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__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $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__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $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_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

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 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) @ i )
    = ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) ).

%------------------------------------------------------------------------------